Precompact abelian groups and topological annihilators \thanks{2000 Mathematics Subject Classification: 22C05 54A20 54D60 (06A15 22A05 54H11)}
For a compact Hausdorff abelian group $K$ and its subgroup $H\leq K$, one defines the {\em $\mathfrak{g}$-closure} $\mathfrak{g}_K(H)$ of $H$ in $K$ as the subgroup consisting of $\chi \in K$ such that $\chi(a_n)\longrightarrow 0$ in $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for every sequence $\{a_n\}$ in $\hat K$ (the Pontryagin dual of $K$) that converges to $0$ in the topology that $H$ induces on $\hat K$. We prove that every countable subgroup of a compact Hausdorff group is $\mathfrak{g}$-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every $\mathfrak{g}$-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator $\mathfrak g$ that coincides with the $G_\delta$-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.
diagxy.tex
 \def\dated#1{\def\thedate{#1}}
 \dated{2003-10-24}
 % This is a package of commutative diagram macros built on top of Xy-pic
 % by Michael Barr (email:  barr@barrs.org).  Its use is unrestricted.  It
 % may be freely distributed, unchanged, for non-commercial or commercial
 % use.  If changed, it must be renamed.  Inclusion in a commercial
 % software package is also permitted, but I would appreciate receiving a
 % free copy for my personal examination and use.  There are no guarantees
 % that this package is good for anything.  I have tested it with LaTeX 2e,
 % LaTeX 2.09 and Plain TeX.  Although I know of no reason it will not work
 % with AMSTeX, I have not tested it.
 % Added 2003-05-10: I now know that the only clash is with \square in
 % amssymb, which should therefore be loaded first.

\newcount\atcode \atcode=\catcode`\@%
\catcode`\@=12
\input xy
\xyoption{arrow}

\newdir{ >}{{ }*!/-.9em/@{>}}
\newdir{ (}{{ }*!/-.5em/@{(}}
\newdir^{ (}{{ }*!/-.5em/@^{(}}
\newdir{< }{!/.9em/@{<}*{ }}

\newdimen\high%
\newdimen\ul%
\newcount\deltax%
\newcount\deltay%
\newcount\deltaX%
\newcount\deltaY%

\newdimen\wdth
\newcount\xend%
\newcount\yend%
\newcount\Xend
\newcount\Yend
\newcount\xpos%
\newcount\ypos%
\newcount\default \default=500%
\newcount\defaultmargin \defaultmargin=150
\newcount\topw%
\newcount\botw%
\newcount\Xpos%
\newcount\Ypos%
\def\ratchet#1#2{\ifnum#1<#2\global #1=#2\fi}%

\catcode`\@=11
\expandafter\ifx\csname @ifnextchar\endcsname\relax
\def\ifnextchar#1#2#3{\let\@tempe
#1\def\@tempa{#2}\def\@tempb{#3}\futurelet
    \@tempc\@ifnch}%
\def\@ifnch{\ifx \@tempc \@sptoken \let\@tempd\@xifnch
      \else \ifx \@tempc \@tempe\let\@tempd\@tempa\else\let\@tempd\@tempb\fi
      \fi \@tempd}%
\def\:{\let\@sptoken= } \:  % this makes \@sptoken a space token
\def\:{\@xifnch} \expandafter\def\: {\futurelet\@tempc\@ifnch}%
\else
\let\ifnextchar\@ifnextchar
\fi
\ifx\check@mathfonts\undefined
\else \check@mathfonts
\fi
\newdimen\axis \axis=\fontdimen22\textfont2
\ul=.01em%
\X@xbase =.01em%
\Y@ybase =.01em%
\def\scalefactor#1{\ul=#1\ul \X@xbase=#1\X@xbase \Y@ybase=#1\Y@ybase}%
\catcode`\@=12%

\def\fontscale#1{%
\if#1h\relax
\font\xydashfont=xydash10 scaled \magstephalf
\font\xyatipfont=xyatip10 scaled \magstephalf
\font\xybtipfont=xybtip10 scaled \magstephalf
\font\xybsqlfont=xybsql10 scaled \magstephalf
\font\xycircfont=xycirc10 scaled \magstephalf
\else
\font\xydashfont=xydash10 scaled \magstep#1%
\font\xyatipfont=xyatip10 scaled \magstep#1%
\font\xybtipfont=xybtip10 scaled \magstep#1%
\font\xybsqlfont=xybsql10 scaled \magstep#1%
\font\xycircfont=xycirc10 scaled \magstep#1%
\fi}

\def\bfig{\vcenter\bgroup\xy}
\def\efig{\endxy\egroup}

\def\car#1#2\nil{#1}%

\def\morphism{\ifnextchar({\morphismp}{\morphismp(0,0)}}%
\def\morphismp(#1){\ifnextchar|{\morphismpp(#1)}{\morphismpp(#1)|a|}}%
\def\morphismpp(#1)|#2|{\ifnextchar/{\morphismppp(#1)|#2|}%
    {\morphismppp(#1)|#2|/>/}}%
\def\morphismppp(#1)|#2|/#3/{%
    \ifnextchar<{\morphismpppp(#1)|#2|/#3/}%
    {\morphismpppp(#1)|#2|/#3/<\default,0>}}%

\def\morphismpppp(#1,#2)|#3|/#4/<#5,#6>[#7`#8;#9]{%
\xend#1\advance \xend by #5%
\yend#2\advance \yend by #6%
\domorphism(#1,#2)|#3|/#4/<#5,#6>[{#7}`{#8};{#9}]}


\def\domorphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8;#9]{%
 %% Check if arrow arg has an @; then don't add it.
\def\next{\car#4.\nil}%
\if@\next\relax
 \if#3l%
  \ifnum #6>0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3m%
    \setbox0\hbox{$#9$}%
   \ifdim \wd0=0pt
     \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4 (\xend,\yend)*+!!<0ex,\axis>{#8}%
   \else
     \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4|-*+<1pt,4pt>{\labelstyle#9}
       (\xend,\yend)*+!!<0ex,\axis>{#8}%
   \fi
 \else \if#3r%
  \ifnum #6<0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3a%
  \ifnum #5>0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3b%
  \ifnum #5<0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar#4 (\xend,\yend)*+!!<0ex,\axis>{#8}%
 \fi\fi\fi\fi\fi%
 %%% Otherwise, have to add @{..}
\else%
 \if#3l%
  \ifnum #6>0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3m%
    \setbox0\hbox{$#9$}%
   \ifdim \wd0=0pt
     \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4} (\xend,\yend)*+!!<0ex,\axis>{#8}%
   \else
     \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}|-*+<1pt,4pt>{\labelstyle#9}
         (\xend,\yend)*+!!<0ex,\axis>{#8}%
   \fi
 \else \if#3r%
  \ifnum #6<0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3a%
  \ifnum #5>0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else \if#3b%
  \ifnum #5<0%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}^-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \else%
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4}_-{#9} (\xend,\yend)*+!!<0ex,\axis>{#8}%
  \fi%
 \else
   \POS(#1,#2)*+!!<0ex,\axis>{#7}\ar@{#4} (\xend,\yend)*+!!<0ex,\axis>{#8}%
 \fi\fi\fi\fi\fi
\fi\ignorespaces}%

\def\vector(#1,#2)/#3/<#4,#5>{%
 \xend#1 \yend#2 \advance\xend by #4 \advance\yend by #5
     \POS(#1,#2)\ar#3 (\xend,\yend)}

\def\squarepppp(#1,#2)|#3|/#4`#5`#6`#7/<#8>[#9]{%
\xpos#1\ypos#2%
\def\next|##1##2##3##4|{%
 \def\xa{##1}\def\xb{##2}\def\xc{##3}\def\xd{##4}\ignorespaces}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#8>%
\def\next[##1`##2`##3`##4;##5`##6`##7`##8]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}\def\noded{##4}%
    \def\labela{##5}\def\labelb{##6}\def\labelc{##7}\def\labeld{##8}\ignorespaces}%
\next[#9]%
\morphism(\xpos,\ypos)|\xd|/{#7}/<\deltax,0>[\nodec`\noded;\labeld]%
\advance \ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/{#5}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\morphism(\xpos,\ypos)|\xa|/{#4}/<\deltax,0>[\nodea`\nodeb;\labela]%
 \advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/{#6}/<0,-\deltay>[\nodeb`\noded;\labelc]%
\ignorespaces}%

\def\square{\ifnextchar({\squarep}{\squarep(0,0)}}%
\def\squarep(#1){\ifnextchar|{\squarepp(#1)}{\squarepp(#1)|alrb|}}%
\def\squarepp(#1)|#2|{\ifnextchar/{\squareppp(#1)|#2|}%
    {\squareppp(#1)|#2|/>`>`>`>/}}%
\def\squareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\squarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\squarepppp(#1)|#2|/#3`#4`#5`#6/<\default,\default>}}%

\def\ptrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#4}/<\deltax,0>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#5}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/{#6}/<-\deltax,-\deltay>[\nodeb`\nodec;\labelc]%
\ignorespaces}%

\def\qtrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#4}/<\deltax,0>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#5}/<\deltax,-\deltay>[\nodea`\nodec;\labelb]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/{#6}/<0,-\deltay>[\nodeb`\nodec;\labelc]%
\ignorespaces}%

\def\dtrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\morphism(\xpos,\ypos)|\xc|/{#6}/<\deltax,0>[\nodeb`\nodec;\labelc]%
\advance\ypos by \deltay\advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/{#4}/<-\deltax,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#5}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\ignorespaces}%

\def\btrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\morphism(\xpos,\ypos)|\xc|/{#6}/<\deltax,0>[\nodeb`\nodec;\labelc]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#4}/<0,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#5}/<\deltax,-\deltay>[\nodea`\nodec;\labelb]%
\ignorespaces}%

\def\Atrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\multiply\deltax by 2%
\morphism(\xpos,\ypos)|\xc|/{#6}/<\deltax,0>[\nodeb`\nodec;\labelc]%
\divide\deltax by 2
\advance\ypos by \deltay\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/{#4}/<-\deltax,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#5}/<\deltax,-\deltay>[\nodea`\nodec;\labelb]%
\ignorespaces}%

\def\Vtrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/{#5}/<\deltax,-\deltay>[\nodea`\nodec;\labelb]%
\multiply\deltax by 2%
\morphism(\xpos,\ypos)|\xa|/{#4}/<\deltax,0>[\nodea`\nodeb;\labela]%
\advance\xpos by \deltax \divide \deltax by 2
\morphism(\xpos,\ypos)|\xc|/{#6}/<-\deltax,-\deltay>[\nodeb`\nodec;\labelc]%
\ignorespaces}%

\def\Ctrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\advance \ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/{#6}/<\deltax,-\deltay>[\nodeb`\nodec;\labelc]%
\advance\ypos by \deltay \advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/{#4}/<-\deltax,-\deltay>[\nodea`\nodeb;\labela]%
\multiply\deltay by 2%
\morphism(\xpos,\ypos)|\xb|/{#5}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\ignorespaces}%

\def\Dtrianglepppp(#1,#2)|#3|/#4`#5`#6/<#7>[#8]{%
\xpos#1\ypos#2%
\def\next|##1##2##3|{\def\xa{##1}\def\xb{##2}\def\xc{##3}}%
\next|#3|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#7>%
\def\next[##1`##2`##3;##4`##5`##6]{%
    \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
    \def\labela{##4}\def\labelb{##5}\def\labelc{##6}}%
\next[#8]%
\advance\xpos by \deltax \advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/{#6}/<-\deltax,-\deltay>[\nodeb`\nodec;\labelc]%
\advance\xpos by -\deltax \advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/{#5}/<\deltax,-\deltay>[\nodea`\nodeb;\labelb]%
\multiply \deltay by 2%
\morphism(\xpos,\ypos)|\xa|/{#4}/<0,-\deltay>[\nodea`\nodec;\labela]%
\ignorespaces}%

\def\ptriangle{\ifnextchar({\ptrianglep}{\ptrianglep(0,0)}}%
\def\ptrianglep(#1){\ifnextchar|{\ptrianglepp(#1)}{\ptrianglepp(#1)|alr|}}%
\def\ptrianglepp(#1)|#2|{\ifnextchar/{\ptriangleppp(#1)|#2|}%
    {\ptriangleppp(#1)|#2|/>`>`>/}}%
\def\ptriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\ptrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\ptrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\qtriangle{\ifnextchar({\qtrianglep}{\qtrianglep(0,0)}}%
\def\qtrianglep(#1){\ifnextchar|{\qtrianglepp(#1)}{\qtrianglepp(#1)|alr|}}%
\def\qtrianglepp(#1)|#2|{\ifnextchar/{\qtriangleppp(#1)|#2|}%
    {\qtriangleppp(#1)|#2|/>`>`>/}}%
\def\qtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\qtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\qtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\dtriangle{\ifnextchar({\dtrianglep}{\dtrianglep(0,0)}}%
\def\dtrianglep(#1){\ifnextchar|{\dtrianglepp(#1)}{\dtrianglepp(#1)|lrb|}}%
\def\dtrianglepp(#1)|#2|{\ifnextchar/{\dtriangleppp(#1)|#2|}%
    {\dtriangleppp(#1)|#2|/>`>`>/}}%
\def\dtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\dtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\btriangle{\ifnextchar({\btrianglep}{\btrianglep(0,0)}}%
\def\btrianglep(#1){\ifnextchar|{\btrianglepp(#1)}{\btrianglepp(#1)|lrb|}}%
\def\btrianglepp(#1)|#2|{\ifnextchar/{\btriangleppp(#1)|#2|}%
    {\btriangleppp(#1)|#2|/>`>`>/}}%
\def\btriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\btrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\btrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Atriangle{\ifnextchar({\Atrianglep}{\Atrianglep(0,0)}}%
\def\Atrianglep(#1){\ifnextchar|{\Atrianglepp(#1)}{\Atrianglepp(#1)|lrb|}}%
\def\Atrianglepp(#1)|#2|{\ifnextchar/{\Atriangleppp(#1)|#2|}%
    {\Atriangleppp(#1)|#2|/>`>`>/}}%
\def\Atriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Atrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Atrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Vtriangle{\ifnextchar({\Vtrianglep}{\Vtrianglep(0,0)}}%
\def\Vtrianglep(#1){\ifnextchar|{\Vtrianglepp(#1)}{\Vtrianglepp(#1)|alb|}}%
\def\Vtrianglepp(#1)|#2|{\ifnextchar/{\Vtriangleppp(#1)|#2|}%
    {\Vtriangleppp(#1)|#2|/>`>`>/}}%
\def\Vtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Vtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Vtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Ctriangle{\ifnextchar({\Ctrianglep}{\Ctrianglep(0,0)}}%
\def\Ctrianglep(#1){\ifnextchar|{\Ctrianglepp(#1)}{\Ctrianglepp(#1)|arb|}}%
\def\Ctrianglepp(#1)|#2|{\ifnextchar/{\Ctriangleppp(#1)|#2|}%
    {\Ctriangleppp(#1)|#2|/>`>`>/}}%
\def\Ctriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Ctrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Ctrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Dtriangle{\ifnextchar({\Dtrianglep}{\Dtrianglep(0,0)}}%
\def\Dtrianglep(#1){\ifnextchar|{\Dtrianglepp(#1)}{\Dtrianglepp(#1)|alb|}}%
\def\Dtrianglepp(#1)|#2|{\ifnextchar/{\Dtriangleppp(#1)|#2|}%
    {\Dtriangleppp(#1)|#2|/>`>`>/}}%
\def\Dtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Dtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%


\def\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\def\next(##1,##2){\xpos##1\ypos##2}%
\next(#1)%
\def\next|##1##2##3##4##5|{\def\xa{##1}\def\xb{##2}%
\def\xc{##3}\def\xd{##4}\def\xe{##5}}%
\next|#2|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#8>%
\def\next[##1`##2`##3`##4;##5`##6`##7`##8`##9]{%
 \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}\def\noded{##4}%
 \def\labela{##5}\def\labelb{##6}\def\labelc{##7}\def\labeld{##8}\def\labele{##9}}%
\next[#9]%
\morphism(\xpos,\ypos)|\xd|/{#6}/<\deltax,0>[\nodeb`\nodec;\labeld]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/{#7}/<\deltax,0>[\nodec`\noded;\labele]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#3}/<-\deltax,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#4}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\morphism(\xpos,\ypos)|\xc|/{#5}/<\deltax,-\deltay>[\nodea`\noded;\labelc]%
\ignorespaces}%

\def\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\def\next(##1,##2){\xpos##1\ypos##2}%
\next(#1)%
\def\next|##1##2##3##4##5|{\def\xa{##1}\def\xb{##2}%
\def\xc{##3}\def\xd{##4}\def\xe{##5}}%
\next|#2|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#8>%
\def\next[##1`##2`##3`##4;##5`##6`##7`##8`##9]{%
 \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}\def\noded{##4}%
 \def\labela{##5}\def\labelb{##6}\def\labelc{##7}\def\labeld{##8}\def\labele{##9}}%
\next[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#3}/<\deltax,0>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xc|/{#5}/<\deltax,-\deltay>[\nodea`\noded;\labelc]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xb|/{#4}/<\deltax,0>[\nodeb`\nodec;\labelb]%
\morphism(\xpos,\ypos)|\xd|/{#6}/<0,-\deltay>[\nodeb`\noded;\labeld]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/{#7}/<-\deltax,-\deltay>[\nodec`\noded;\labele]%
\ignorespaces}%

\def\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\def\next(##1,##2){\xpos##1\ypos##2}%
\next(#1)%
\def\next|##1##2##3##4##5|{\def\xa{##1}\def\xb{##2}%
\def\xc{##3}\def\xd{##4}\def\xe{##5}}%
\next|#2|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#8>%
\def\next[##1`##2`##3`##4;##5`##6`##7`##8`##9]{%
 \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}\def\noded{##4}%
 \def\labela{##5}\def\labelb{##6}\def\labelc{##7}\def\labeld{##8}\def\labele{##9}}%
\next[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xe|/{#7}/<0,-\deltay>[\nodec`\noded;\labele]%
\advance\xpos by -\deltax%
\morphism(\xpos,\ypos)|\xc|/{#5}/<\deltax,0>[\nodeb`\nodec;\labelc]%
\morphism(\xpos,\ypos)|\xd|/{#6}/<\deltax,-\deltay>[\nodeb`\noded;\labeld]%
\advance\ypos by \deltay%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/{#3}/<-\deltax,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#4}/<0,-\deltay>[\nodea`\nodec;\labelb]%
\ignorespaces}%

\def\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\def\next(##1,##2){\xpos##1\ypos##2}%
\next(#1)%
\def\next|##1##2##3##4##5|{\def\xa{##1}\def\xb{##2}%
\def\xc{##3}\def\xd{##4}\def\xe{##5}}%
\next|#2|%
\def\next<##1,##2>{\deltax=##1\deltay=##2\ignorespaces}%
\next<#8>%
\def\next[##1`##2`##3`##4;##5`##6`##7`##8`##9]{%
 \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}\def\noded{##4}%
 \def\labela{##5}\def\labelb{##6}\def\labelc{##7}\def\labeld{##8}\def\labele{##9}}%
\next[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/{#5}/<\deltax,0>[\nodeb`\nodec;\labelc]%
\morphism(\xpos,\ypos)|\xd|/{#6}/<0,-\deltay>[\nodeb`\noded;\labeld]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/{#3}/<0,-\deltay>[\nodea`\nodeb;\labela]%
\morphism(\xpos,\ypos)|\xb|/{#4}/<\deltax,-\deltay>[\nodea`\nodec;\labelb]%
\advance\ypos by -\deltay%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/{#7}/<-\deltax,-\deltay>[\nodec`\noded;\labele]%
\ignorespaces}%

\def\Atrianglepair{\ifnextchar({\Atrianglepairp}{\Atrianglepairp(0,0)}}%
\def\Atrianglepairp(#1){\ifnextchar|{\Atrianglepairpp(#1)}%
{\Atrianglepairpp(#1)|lmrbb|}}%
\def\Atrianglepairpp(#1)|#2|{\ifnextchar/{\Atrianglepairppp(#1)|#2|}%
    {\Atrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Atrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Vtrianglepair{\ifnextchar({\Vtrianglepairp}{\Vtrianglepairp(0,0)}}%
\def\Vtrianglepairp(#1){\ifnextchar|{\Vtrianglepairpp(#1)}%
{\Vtrianglepairpp(#1)|aalmr|}}%
\def\Vtrianglepairpp(#1)|#2|{\ifnextchar/{\Vtrianglepairppp(#1)|#2|}%
    {\Vtrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Vtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Ctrianglepair{\ifnextchar({\Ctrianglepairp}{\Ctrianglepairp(0,0)}}%
\def\Ctrianglepairp(#1){\ifnextchar|{\Ctrianglepairpp(#1)}%
{\Ctrianglepairpp(#1)|lrmlr|}}%
\def\Ctrianglepairpp(#1)|#2|{\ifnextchar/{\Ctrianglepairppp(#1)|#2|}%
    {\Ctrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Ctrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Dtrianglepair{\ifnextchar({\Dtrianglepairp}{\Dtrianglepairp(0,0)}}%
\def\Dtrianglepairp(#1){\ifnextchar|{\Dtrianglepairpp(#1)}%
{\Dtrianglepairpp(#1)|lrmlr|}}%
\def\Dtrianglepairpp(#1)|#2|{\ifnextchar/{\Dtrianglepairppp(#1)|#2|}%
    {\Dtrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Dtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\pplace[#1](#2,#3)[#4]{\POS(#2,#3)*+!!<0ex,\axis>!#1{#4}\ignorespaces}%
\def\cplace(#1,#2)[#3]{\POS(#1,#2)*+!!<0ex,\axis>{#3}\ignorespaces}%
\def\place{\ifnextchar[{\pplace}{\cplace}}

\def\pullback#1]#2]{\square#1]\trident#2]\ignorespaces}%

\def\tridentppp|#1#2#3|/#4`#5`#6/<#7,#8>[#9]{%
\def\next[##1;##2`##3`##4]{\def\nodee{##1}\def\labele{##2}%
   \def\labelf{##3}\def\labelg{##4}}%
\next[#9]%
\advance \xpos by -\deltax%
\advance \xpos by -#7\advance \ypos by #8%
\advance\deltax by #7%
\morphism(\xpos,\ypos)|#1|/{#4}/<\deltax,-#8>[\nodee`\nodeb;\labele]%
\advance\deltax by -#7%
\morphism(\xpos,\ypos)|#2|/{#5}/<#7,-#8>[\nodee`\nodea;\labelf]%
\advance\deltay by #8%
\morphism(\xpos,\ypos)|#3|/{#6}/<#7,-\deltay>[\nodee`\nodec;\labelg]%
\ignorespaces}%

\def\trident{\ifnextchar|{\tridentp}{\tridentp|amb|}}%
\def\tridentp|#1|{\ifnextchar/{\tridentpp|#1|}{\tridentpp|#1|/{>}`{>}`{>}/}}%
\def\tridentpp|#1|/#2/{\ifnextchar<{\tridentppp|#1|/#2/}%
  {\tridentppp|#1|/#2/<500,500>}}%

\def\setmorphismwidth#1#2#3#4{%
 \setbox0=\hbox{$#1{\labelstyle#3#3}#2$}#4=\wd0%
 \divide #4 by 2 \divide #4 by \ul%
 \advance #4 by 350 \ratchet{#4}{500}}%

\def\setSquarewidth[#1`#2`#3`#4;#5`#6`#7`#8]{%
 \setmorphismwidth{#1}{#2}{#5}{\topw}%
 \setmorphismwidth{#3}{#4}{#8}{\botw}%
\ratchet{\topw}{\botw}}%

\def\Squarepppp(#1)|#2|/#3/<#4>[#5]{%
 \setSquarewidth[#5]%
 \squarepppp(#1)|#2|/#3/<\topw,#4>[#5]%
\ignorespaces}%

\def\Square{\ifnextchar({\Squarep}{\Squarep(0,0)}}%
\def\Squarep(#1){\ifnextchar|{\Squarepp(#1)}{\Squarepp(#1)|alrb|}}%
\def\Squarepp(#1)|#2|{\ifnextchar/{\Squareppp(#1)|#2|}%
    {\Squareppp(#1)|#2|/>`>`>`>/}}%
\def\Squareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\Squarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\Squarepppp(#1)|#2|/#3`#4`#5`#6/<\default>}}%

\def\hSquarespppp(#1,#2)|#3|/#4/<#5>[#6;#7]{%
\Xpos=#1\Ypos=#2%
\def\next|##1##2##3##4##5##6##7|{%
 \def\Xa{##1}\def\Xb{##2}\def\Xc{##3}\def\Xd{##4}%
 \def\Xe{##5}\def\Xf{##6}\def\Xg{##7}}%
\next|#3|%
\deltaY=#5%
\def\next[##1`##2`##3`##4`##5`##6]{%
 \def\Nodea{##1}\def\Nodeb{##2}\def\Nodec{##3}%
 \def\Noded{##4}\def\Nodee{##5}\def\Nodef{##6}}%
\next[#6]%
\def\next[##1`##2`##3`##4`##5`##6`##7]{%
 \def\Labela{##1}\def\Labelb{##2}\def\Labelc{##3}\def\Labeld{##4}%
 \def\Labele{##5}\def\Labelf{##6}\def\Labelg{##7}}%
\next[#7]%
\dohSquares/#4/}%

\def\dohSquares/#1`#2`#3`#4`#5`#6`#7/{%
\Squarepppp(\Xpos,\Ypos)|\Xa\Xc\Xd\Xf|/#1`#3`#4`#6/<\deltaY>%
 [\Nodea`\Nodeb`\Noded`\Nodee;\Labela`\Labelc`\Labeld`\Labelf]%
 \advance \Xpos by \topw
\Squarepppp(\Xpos,\Ypos)|\Xb\Xd\Xe\Xg|/#2``#5`#7/<\deltaY>%
[\Nodeb`\Nodec`\Nodee`\Nodef;\Labelb``\Labele`\Labelg]%
\ignorespaces}%

\def\hSquares{\ifnextchar({\hSquaresp}{\hSquaresp(0,0)}}%
\def\hSquaresp(#1){\ifnextchar|{\hSquarespp(#1)}{\hSquarespp%
(#1)|aalmrbb|}}%
\def\hSquarespp(#1)|#2|{\ifnextchar/{\hSquaresppp(#1)|#2|}%
    {\hSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}%
\def\hSquaresppp(#1)|#2|/#3/{%
    \ifnextchar<{\hSquarespppp(#1)|#2|/#3/}%
    {\hSquarespppp(#1)|#2|/#3/<\default>}}%

\def\vSquarespppp(#1,#2)|#3|/#4/<#5,#6>[#7;#8]{%
\Xpos=#1\Ypos=#2%
\def\next|##1##2##3##4##5##6##7|{%
 \def\Xa{##1}\def\Xb{##2}\def\Xc{##3}\def\Xd{##4}%
 \def\Xe{##5}\def\Xf{##6}\def\Xg{##7}}%
\next|#3|%
\deltaX=#5%
\deltaY=#6%
\def\next[##1`##2`##3`##4`##5`##6]{%
 \def\Nodea{##1}\def\Nodeb{##2}\def\Nodec{##3}%
 \def\Noded{##4}\def\Nodee{##5}\def\Nodef{##6}}%
\next[#7]%
\def\next[##1`##2`##3`##4`##5`##6`##7]{%
 \def\Labela{##1}\def\Labelb{##2}\def\Labelc{##3}\def\Labeld{##4}%
 \def\Labele{##5}\def\Labelf{##6}\def\Labelg{##7}}%
\next[#8]%
\dovSquares/#4/\ignorespaces}%

\def\dovSquares/#1`#2`#3`#4`#5`#6`#7/{%
\setmorphismwidth{\Nodea}{\Nodeb}{\Labela}{\topw}%
\setmorphismwidth{\Nodec}{\Noded}{\Labeld}{\botw}%
\ratchet{\topw}{\botw}%
\setmorphismwidth{\Nodee}{\Nodef}{\Labelg}{\botw}%
\ratchet{\topw}{\botw}%
\square(\Xpos,\Ypos)|\Xd\Xe\Xf\Xg|/`#5`#6`#7/<\topw,\deltaX>%
 [\Nodec`\Noded`\Nodee`\Nodef;`\Labele`\Labelf`\Labelg]%
\advance \Ypos by \deltaX%
\square(\Xpos,\Ypos)|\Xa\Xb\Xc\Xd|/#1`#2`#3`#4/<\topw,\deltaY>%
 [\Nodea`\Nodeb`\Nodec`\Noded;\Labela`\Labelb`\Labelc`\Labeld]%
}%

\def\vSquares{\ifnextchar({\vSquaresp}{\vSquaresp(0,0)}}%
\def\vSquaresp(#1){\ifnextchar|{\vSquarespp(#1)}{\vSquarespp%
(#1)|alrmlrb|}}%
\def\vSquarespp(#1)|#2|{\ifnextchar/{\vSquaresppp(#1)|#2|}%
    {\vSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}%
\def\vSquaresppp(#1)|#2|/#3/{%
    \ifnextchar<{\vSquarespppp(#1)|#2|/#3/}%
    {\vSquarespppp(#1)|#2|/#3/<\default,\default>}}%

\def\osquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{\squarepppp%
 (#1)|#2|/#3`#4`#5`#6/<#7>[#8]%
 \let\Nodea\nodea\let\Nodeb\nodeb%
\let\Nodec\nodec\let\Noded\noded\Xpos=\xpos\Ypos=\ypos%
\deltaX=\deltax \deltaY=\deltay \isquare}

\def\cube{\ifnextchar({\osquarep}{\osquarep(0,0)}}%
\def\osquarep(#1){\ifnextchar|{\osquarepp(#1)}{\osquarepp(#1)|alrb|}}%
\def\osquarepp(#1)|#2|{\ifnextchar/{\osquareppp(#1)|#2|}%
    {\osquareppp(#1)|#2|/>`>`>`>/}}%
\def\osquareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\osquarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\osquarepppp(#1)|#2|/#3`#4`#5`#6/<1500,1500>}}%

\def\isquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{%
 \squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]%
\ifnextchar|{\cubep}{\cubep|mmmm|}}%
\def\cubep|#1|{\ifnextchar/{\cubepp|#1|}{\cubepp|#1|/>`>`>`>/}}%

\def\isquare{\ifnextchar({\isquarep}{\isquarep(\default,\default)}}%
\def\isquarep(#1){\ifnextchar|{\isquarepp(#1)}{\isquarepp(#1)|alrb|}}
\def\isquarepp(#1)|#2|{\ifnextchar/{\isquareppp(#1)|#2|}%
    {\isquareppp(#1)|#2|/>`>`>`>/}}%
\def\isquareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\isquarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\isquarepppp(#1)|#2|/#3`#4`#5`#6/<500,500>}}%

\def\cubepp|#1#2#3#4|/#5`#6`#7`#8/[#9]{%
\def\next[##1`##2`##3`##4]{\gdef\Labela{##1}%
\gdef\Labelb{##2}\gdef\Labelc{##3}\gdef\Labeld{##4}}\next[#9]%
\xend\xpos \yend\ypos
\Xend\xend\advance\Xend by -\Xpos
\Yend\yend\advance\Yend by -\Ypos
\domorphism(\Xpos,\Ypos)|#2|/#6/<\Xend,\Yend>[\Nodeb`\nodeb;\Labelb]%
\advance\Xpos by-\deltaX
\advance\xend by-\deltax
\Xend\xend\advance\Xend by -\Xpos
\domorphism(\Xpos,\Ypos)|#1|/#5/<\Xend,\Yend>[\Nodea`\nodea;\Labela]%
\advance\Ypos by-\deltaY
\advance\yend by-\deltay
\Yend\yend\advance\Yend by -\Ypos
\domorphism(\Xpos,\Ypos)|#3|/#7/<\Xend,\Yend>[\Nodec`\nodec;\Labelc]%
\advance\Xpos by\deltaX
\advance\xend by\deltax
\Xend\xend\advance\Xend by -\Xpos
\domorphism(\Xpos,\Ypos)|#4|/#8/<\Xend,\Yend>[\Noded`\noded;\Labeld]%
\ignorespaces}

\def\setwdth#1#2{\setbox0\hbox{$\labelstyle#1$}\wdth=\wd0
\setbox0\hbox{$\labelstyle#2$}\ifnum\wdth<\wd0 \wdth=\wd0 \fi}

\def\topppp/#1/<#2>^#3_#4{\allowbreak\mathrel{%
\ifnum#2=0
   \setwdth{#3}{#4}\deltax=\wdth \divide \deltax by \ul
   \advance \deltax by \defaultmargin  \ratchet{\deltax}{200}%
\else \deltax #2
\fi
\xy\ar@{#1}^{#3}_{#4}(\deltax,0) \endxy
\ignorespaces}}

\def\toppp/#1/<#2>^#3{\ifnextchar_{\topppp/#1/<#2>^{#3}}{\topppp/#1/<#2>^{#3}_{}}}
\def\topp/#1/<#2>{\ifnextchar^{\toppp/#1/<#2>}{\toppp/#1/<#2>^{}}}
\def\toop/#1/{\ifnextchar<{\topp/#1/}{\topp/#1/<0>}}
\def\to{\ifnextchar/{\toop}{\toop/>/}}


\def\mon{\to/ >->/}
\def\epi{\to/->>/}
\def\toleft{\to/<-/}
\def\monleft{\to/<-< /}
\def\epileft{\to/<<-/}


\def\twopppp/#1`#2/<#3>^#4_#5{\allowbreak\mathrel{%
\ifnum0=#3
  \setwdth{#4}{#5}\deltax=\wdth \divide \deltax by \ul \advance \deltax
  by \defaultmargin \ratchet{\deltax}{200}%
\else \deltax#3 \fi
\xy\ar@{#1}@<2.5pt>^{#4}(\deltax,0)%
\ar@{#2}@<-2.5pt>_{#5}(\deltax,0)\endxy\ignorespaces}}

\def\twoppp/#1`#2/<#3>^#4{\ifnextchar_{\twopppp/#1`#2/<#3>^{#4}}%
  {\twopppp/#1`#2/<#3>^{#4}_{}}}
\def\twopp/#1`#2/<#3>{\ifnextchar^{\twoppp/#1`#2/<#3>}{\twoppp/#1`#2/<#3>^{}}}
\def\twop/#1`#2/{\ifnextchar<{\twopp/#1`#2/}{\twopp/#1`#2/<0>}}
\def\two{\ifnextchar/{\twop}{\twop/>`>/}}

\def\twoleft{\two/<-`<-/}

\def\threeppppp/#1`#2`#3/<#4>^#5|#6_#7{\allowbreak\mathrel{%
\ifnum0=#4
\setbox0\hbox{$\labelstyle#5$}\wdth=\wd0
\setbox0\hbox{$\labelstyle#6$}\ifnum\wdth<\wd0 \wdth=\wd0 \fi
\setbox0\hbox{$\labelstyle#7$}\ifnum\wdth<\wd0 \wdth=\wd0 \fi
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\def\threep/#1`#2`#3/{\ifnextchar<{\threepp/#1`#2`#3/}%
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\def\three{\ifnextchar/{\threep}{\threep/>`>`>/}}


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 \deltax#1\deltay#2%
 \deltaX=\ifnum\deltax<0-\fi\deltax
 \deltaY=\ifnum\deltay<0-\fi\deltay
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 \advance \ypos by \Ypos
 \xy \ar@{=>}(\xpos,\ypos) \endxy
}\ignorespaces}

\def\iiixiiipppppp(#1,#2)|#3|/#4/<#5>#6<#7>[#8;#9]{%
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 \def\next|##1##2##3##4##5##6##7|{\def\xa{##1}\def\xb{##2}%
 \def\xc{##3}\def\xd{##4}\def\xe{##5}\def\xf{##6}\nextt|##7|}%
 \def\nextt|##1##2##3##4##5##6|{\def\xg{##1}\def\xh{##2}%
 \def\xi{##3}\def\xj{##4}\def\xk{##5}\def\xl{##6}}%
 \next|#3|%
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 \ifodd\topw \def\zb{}\else\def\zb{\relax}\fi \divide\topw by 2
 \ifodd\topw \def\za{}\else\def\za{\relax}\fi}%
 \next{#6}%
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 \def\noded{##4}\def\nodee{##5}\def\nodef{##6}%
 \def\nodeg{##7}\def\nodeh{##8}\def\nodei{##9}}%
 \next[#8]%
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 \def\labela{##1}\def\labelb{##2}\def\labelc{##3}%
 \def\labeld{##4}\def\labele{##5}\def\labelf{##6}\nextt[##7]}%
 \def\nextt[##1`##2`##3`##4`##5`##6]{%
 \def\labelg{##1}\def\labelh{##2}\def\labeli{##3}%
 \def\labelj{##4}\def\labelk{##5}\def\labell{##6}}%
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 \advance\xpos by \deltax
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\def\iiixiiip(#1){\ifnextchar|{\iiixiiipp(#1)}%
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   {\iiixiiipppppp(#1)|#2|/#3/<#4>{#5}<400,400>}}%

\def\iiixiipppppp(#1,#2)|#3|/#4/<#5>#6<#7>[#8;#9]{%
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 \def\xc{##3}\def\xd{##4}\def\xe{##5}\def\xf{##6}\def\xg{##7}}%
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 \def\next<##1,##2>{\deltax##1\deltay##2}%
 \next<#5>%
 \deltaX#7
 \topw#6
 \def\next{%
 \ifodd\topw \def\za{}\else\def\za{\relax}\fi \divide\topw by 2
 \ifodd\topw \def\zb{}\else\def\zb{\relax}\fi \divide\topw by 2
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 \ifodd\topw \def\zd{}\else\def\zd{\relax}\fi}%
 \next
 \def\next[##1`##2`##3`##4`##5`##6]{%
 \def\nodea{##1}\def\nodeb{##2}\def\nodec{##3}%
 \def\noded{##4}\def\nodee{##5}\def\nodef{##6}}%
 \next[#8]%
 \def\next[##1`##2`##3`##4`##5`##6`##7]{%
 \def\labela{##1}\def\labelb{##2}\def\labelc{##3}%
 \def\labeld{##4}\def\labele{##5}\def\labelf{##6}\def\labelg{##7}}%
 \next[#9]%
 \def\next/##1`##2`##3`##4`##5`##6`##7/{%
 \ifx\zc\empty\relax\morphism(\xpos,\ypos)<\deltaX,0>[0`\noded;]\fi
 \advance\xpos by\deltaX
 \morphism(\xpos,\ypos)|\xc|/##3/<\deltax,0>[\noded`\nodee;\labelc]%
 \advance\xpos by \deltax
 \morphism(\xpos,\ypos)|\xd|/##4/<\deltax,0>[\nodee`\nodef;\labeld]%
 \advance\xpos by \deltax
 \ifx\zd\empty\relax  \morphism(\xpos,\ypos)<\deltaX,0>[\nodef`0;]\fi
 \advance\xpos by -\deltaX  \advance\xpos by -\deltax
 \advance\xpos by -\deltax  \advance\ypos by \deltay
 \ifx\za\empty\relax\morphism(\xpos,\ypos)<\deltaX,0>[0`\nodea;]\fi
 \advance\xpos by\deltaX
 \morphism(\xpos,\ypos)|\xa|/##1/<\deltax,0>[\nodea`\nodeb;\labela]%
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 \ifx\zb\empty\relax \morphism(\xpos,\ypos)<\deltaX,0>[\nodec`0;]\fi}%
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\def\iiixii{\ifnextchar({\iiixiip}{\iiixiip(0,0)}}%
\def\iiixiip(#1){\ifnextchar|{\iiixiipp(#1)}%
  {\iiixiipp(#1)|aabblmr|}}%
\def\iiixiipp(#1)|#2|{\ifnextchar/{\iiixiippp(#1)|#2|}%
    {\iiixiippp(#1)|#2|/>`>`>`>`>`>`>/}}%
\def\iiixiippp(#1)|#2|/#3/{%
    \ifnextchar<{\iiixiipppp(#1)|#2|/#3/}%
    {\iiixiipppp(#1)|#2|/#3/<\default,\default>}}%
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\def\node#1(#2,#3)[#4]{%
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\expandafter\gdef\csname y#1\endcsname{#3}%
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\def\arrowpp|#1|/#2/[#3`#4;#5]{%
\xfinish=\csname x#4\endcsname
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gclose.tex
\documentclass[12pt,letterpaper,twoside]{article}

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\author{G\'abor Luk\'acs
\thanks{I gratefully acknowledge the generous financial support received
from the Alexander von Humboldt Foundation, the Killam Trusts, and 
Dalhousie University that enabled me to do this research.}}
   
\title{Precompact abelian groups and topological annihilators
\thanks{2000 Mathematics Subject Classification: 22C05 54A20 54D60 (06A15 22A05 54H11)}}

\hyphenation{to-po-lo-gies}
\hyphenation{mo-no-mor-phism}
\hyphenation{me-tri-za-ble}


%\renewcommand{\ker}{\operatorname{Ker}}
\providecommand{\KER}{\operatorname{\mathfrak{Ker}}}
\providecommand{\im}{\operatorname{im}}





\begin{document}

\makeatletter
\let\mytitle\@title
\chead{\small\itshape G. Luk\'acs / \mytitle }
\fancyhead[RO,LE]{\small \thepage}
\makeatother

\maketitle

\def\thanks#1{} 

\thispagestyle{empty}


% References that we may or may not mention explicitly:
\nocite{ComfHernTrig}
\nocite{ComfRem} 
\nocite{HernMaca} 
\nocite{Reid} 
\nocite{Varop}

\begin{abstract}
For a compact Hausdorff abelian group $K$ and its subgroup
$H\leq K$, one defines the
{\em $\mathfrak{g}$-closure} $\mathfrak{g}_K(H)$ of $H$ in $K$
as the subgroup consisting of $\chi \in K$ such that 
$\chi(a_n)\longrightarrow 0$ in
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for
every sequence $\{a_n\}$ in $\hat K$
(the Pontryagin dual of $K$) that converges to $0$ in the topology
that $H$ induces on $\hat K$. We prove that every countable subgroup of 
a compact Hausdorff group is $\mathfrak{g}$-closed, and thus give a 
positive answer to two problems of Dikranjan, Milan and Tonolo.
We also show that every $\mathfrak{g}$-closed subgroup
of a compact Hausdorff group is realcompact.
The techniques developed in the paper are used to
construct a close relative of the closure operator $\mathfrak g$ that 
coincides with the $G_\delta$-closure on compact Hausdorff abelian groups, 
and thus captures realcompactness and pseudocompactness of subgroups.
\end{abstract}


\section{Introduction}

\label{sect:intro}

For a sequence $\underline u=(u_n)$ of integers, one defines
$t_{\underline u}(\mathbb T) =\{x \in \mathbb{T} \mid u_n x \longrightarrow 0\}$,
the subgroup of {\em topologically $\underline u$-torsion} elements in
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and sets
$\mathfrak{t}(H)=\bigcap\{t_{\underline u}(\mathbb{T}) \mid
H \leq t_{\underline u}(\mathbb{T}) \}$ for every subgroup
$H \leq \mathbb{T}$ (cf. \cite{Diktoptor}, \cite{BarDikMilWeb}, \cite{DikMilTon}).
This enables one to speak of $\mathfrak{t}$-closed subgroups 
(i.e., $H \leq \mathbb{T}$ such that $\mathfrak{t}(H)=H$). The subgroups
$t_{\underline u}(\mathbb{T})$ are called {\em basic} $\mathfrak{t}$-closed
subgroups.
B\'{\i}r\'o, Deshouillers and  S\'os  proved:

\begin{ftheorem}[{\cite[Theorem~2]{BirDesSos}}] \label{fthm:BirDesSos}
Every countable subgroup of $\,\mathbb{T}$ is basic $\mathfrak{t}$-closed.
\end{ftheorem}

Although it is possible to define $t_{\underline u}(G)$ for an arbitrary
abelian topological group in a similar fashion, the closure $\mathfrak{t}_G$ 
thus obtained turned out to have worse properties than anticipated: 
Dikranjan and Di Santo  \cite{DikSan} showed that if $L$ is a non-discrete 
locally compact Hausdorff abelian group and $L \not\cong \mathbb{T}$, 
then there is $x\in L$ such that the subgroup
$\langle x \rangle$ is not even $\mathfrak{t}$-closed in $L$.
In particular, Theorem~\ref{fthm:BirDesSos} fails 
for $L \not\cong \mathbb{T}$.

Following Dikranjan, Milan and Tonolo \cite[2.1]{DikMilTon}, given an abelian 
topological group $G$ and a sequence $\underline{u}=(u_n)$
in its Pontryagin dual $\hat G$, we put 
$s_{\underline u}(G)=\{x\in G\mid u_n(x)\longrightarrow 0
\text{ in $\mathbb{T}$}\}$, the subgroup of {\em topologically
${\underline u}$-annihilating} elements of $G$. For every
subgroup $H \leq G$, we set 
\begin{equation}
\mathfrak g_G(H)=\bigcap\{ s_{\underline u}(G) \mid 
\underline{u}\in\hat G^\mathbb{N},H \leq s_{\underline u}(G)\}.
\end{equation}
One says that $H\leq G$ is {\em $\mathfrak{g}$-closed}  in $G$ if
$\mathfrak{g}_G(H)=H$, and $H$ is {\em $\mathfrak{g}$-dense} there
if $\mathfrak{g}_G(H)=G$.
Using this terminology, $\mathfrak g$ is the largest closure operator
with respect to which each {\em basic $\mathfrak{g}$-closed} set
$s_{\underline u}(G)$ is closed. (For details on closure operators
of topological groups, we refer the reader to the survey of 
Dikranjan \cite{DikCOTA}.)

%Note that 
%$t_{\underline u}(\mathbb{T})=s_{\underline u}(\mathbb{T})$
%for  every $\underline u \in \mathbb{Z}^\mathbb{N}$, but
%$\underline u \not \in \hat G^\mathbb{N}$ for an arbitrary $G$,
%and so the above-mentioned equality cannot hold in general.




For a category $\mathsf{T}$,  $\mathsf{Grp(T)}$ 
stands for the category of group objects in $\mathsf{T}$ and their group
homomorphisms, and $\mathsf{Ab(T)}$ for its full subcategory consisting
of the abelian group objects.
Thus, $\mathsf{Grp(Top)}$, $\mathsf{Grp(CompH)}$,
$\mathsf{Ab(Top)}$, and $\mathsf{Ab(CompH)}$ are the categories
of topological groups (without separation axioms), compact Hausdorff 
topological groups, and their abelian counterparts, respectively. 

For $\,G \in \mathsf{Grp(Top)}$, we denote by $\rho_G\colon G\rightarrow bG$
the {\em Bohr-compactification} of $G$, that is, 
the reflection of $G$ into $\mathsf{Grp(CompH)}$. One puts $G^+$ for
$G$ equipped with the initial topology induced by $\rho_G$.
The kernel of $\rho_G$ is denoted by $\mathbf{n}(G)$ and called the 
{\em von Neumann radical} of $G$. One says that 
$G$ is {\em maximally almost periodic} if $\mathbf{n}(G)=1$---in
other words, if $\rho_G$ is injective (cf.~\cite{NeuWig}).

\bigskip

In this paper, we study the closure operator $\mathfrak{g}$
(and its relatives that are introduced in section~\ref{sect:cont})
by means of modification of precompact group topologies.
Our main results in the context of the 
$\mathfrak{g}$-closure itself are Theorems 
\ref{thm:positive}-\ref{thm:gc:MAP} below, whose proofs appear in 
section~\ref{sect:gclosure}.

\begin{theorem} \label{thm:positive}
Let $H$ be a countable subgroup of a compact Hausdorff abelian 
topological group $K$. Then $H$ is $\mathfrak{g}$-closed in $K$.
\end{theorem}


Theorem~\ref{thm:positive} is a positive solution for two problems of 
Dikranjan,  Milan and Tonolo \cite[Problem~5.1, Question~5.2]{DikMilTon}, 
and it generalizes Theorem~\ref{fthm:BirDesSos}.
Independently, Dikranjan and Kunen \cite[1.9]{DikKun} 
and Mathias Beiglb\"ock, Christian Steineder and Reinhard Winkler
\cite{BeiSteWin} have also obtained Theorem~\ref{thm:positive}.

\begin{theorem} \label{thm:gc-realcompact}
Every $\mathfrak g$-closed subgroup of a compact Hausdorff
abelian group is realcompact.
\end{theorem}

\begin{theorem} \label{thm:gc:MAP}
For $G\in\mathsf{Ab(Top)}$, the following statements are equivalent:

\begin{list}{{\rm (\roman{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
$G$ is maximally almost periodic;

\item
every countable subgroup of $G$ is $\mathfrak g$-closed;

\item
every cyclic subgroup of $G$ is $\mathfrak g$-closed.

\end{list}
\end{theorem}


Theorem~\ref{thm:gc:MAP} is a variant of the main result of
\cite{DikMilTon}, and so its novelty lies in its significantly simplified 
proof that uses techniques developed below.



\bigskip

One says that $G \in \mathsf{Grp(Top)}$ is {\em precompact} if for every 
neighborhood $U$ of the identity in $G$, there exists a finite subset 
$F\subseteq G$ such that $G=FU$. 

In order to describe  our method of investigating the 
closure operator $\mathfrak{g}$ (and its relatives that 
are introduced in section~\ref{sect:cont}), we first survey
some basic facts related to precompact abelian groups.
Fix $K \in \mathsf{Ab(CompH)}$, and 
let $A=\hat K$ be its Pontryagin dual. It is a 
well-known result of Comfort and Ross \cite{ComfRoss} that every
precompact Hausdorff group topology $\tau$ on $A$ coincides with
the initial topology on $A$ with respect to characters $\chi$ in a 
dense subgroup $H\leq K$. In other words, one has $\tau = \tau_H$,
where $\tau_H$ is the coarsest (group) topology on $A$ such that each
$\chi\colon A \rightarrow \mathbb{T}$ in $H$ is continuous. 
There is a close relationship between properties of $(A,\tau_H)$ and $H$.
To mention only two basic ones, the identity map  
$(A,\tau_{H_1})\rightarrow (A,\tau_{H_2})$ is continuous  (i.e., 
$\tau_{H_1} \supseteq \tau_{H_2}$) if and only if $H_1 \geq H_2$ 
(cf. \cite[2.6]{HernMaca}), and $(A,\tau_H)$ is metrizable if and only 
if $H$ is countable (cf. \cite[2.12]{HernMaca}). For a detailed
account of results of this nature, we refer the reader to \cite{HernMaca} and
\cite{ComfHernTrig}. 




For $H\leq K$, consider
$H^\perp:= \bigcap\limits_{\chi\in H} \ker\chi$. It follows from the
Pontryagin duality that the dual of $A/H^\perp$ is the closure
$\bar H$ of $H$ in $K$, and $H^\perp = \bar H^\perp$. Thus,
although $H$ need not separate points in $A$, it always separates
the points of $A/H^\perp$. In particular, $H$ separates the points 
of $A$ if and only if $H$ is dense 
in $K$ (cf. \cite{ComfRoss}).
Therefore, in the context of characters
and duality, the precompact (but not necessarily Hausdorff) group 
$(A,\tau_H)$ is equivalent to the precompact Hausdorff group
$(A/H^\perp,\tau_H)$, because the latter is the maximal Hausdorff
quotient of the earlier (see \cite[2.10]{HernMaca}). 





The key to our approach is our viewing $\mathfrak{g}$ as a functor
that assigns to each precompact abelian group $(A,\tau_H)$
the group $(A,\tau_{\mathfrak g_{\empty_K}(H)})$. We show that
$\mathfrak g_K(H)$ consists of the elements of $K$ that are 
sequentially continuous as characters of $(A,\tau_H)$
(i.e., $\chi(a_n)\longrightarrow \chi(a_0)$ 
for every sequence $\{a_n\}$ in $A$ such that
$a_n \longrightarrow a_0$ in $\tau_H$; cf. Lemma~\ref{lemma:f}(b)).

\bigskip

In the course of investigating $\mathfrak{g}$, we discovered
that many of its properties remain true if sequences are replaced
with  a ``nice" method of choosing filter-bases in $\hat G$. 
Thus, in order to achieve a greater generality, in
section~\ref{sect:cont}, we study such mutants of 
$\mathfrak{g}$. Each closure operator of this type gives rise to 
a full coreflective subcategory of the category of precompact abelian groups.
In section~\ref{sect:realcompact}, we present such a ``mutant" 
$\mathfrak{l}$ of $\mathfrak{g}$ (in the sense of 
section~\ref{sect:cont}) that coincides with the 
$G_\delta$-closure on compact Hausdorff abelian groups
(cf.~Theorem~\ref{thm:char:l}); thus, 
$\mathfrak{l}$-closed subgroups are the realcompact ones, 
and $\mathfrak{l}$-dense subgroups are the dense pseudocompact ones.
Finally, in section~\ref{sect:gclosure},  the general theory developed in 
section~\ref{sect:cont} is applied to the closure operator $\mathfrak{g}$,
and Theorems~\ref{thm:positive}-\ref{thm:gc:MAP} are proven. 






\section{Continuity types}

\label{sect:cont}

In this section, we develop the filter-base analogue of the 
$\mathfrak{g}$-closure. Let $G \in \mathsf{Ab(Top)}$ and let $\mathcal{F}$ 
be a filter-base on its Pontryagin dual $\hat G$. We put
\begin{equation}
z_{\mathcal{F}}(G) = \{ \chi\in G \mid \chi(\mathcal{F}) \longrightarrow 0 
\text{ in } \mathbb{T} \},
\end{equation}
where elements of $G$ are seen as characters of $\hat G$, and thus 
they are denoted by Greek letters.


\begin{lemma} \label{lemma:fCLOP:funct}
Let $\varphi\colon G_1 \rightarrow G_2$ be a morphism in 
$\mathsf{Ab(Top)}$, and
$\mathcal{F}$ be a filter-base on $\hat G_2$. Then
\begin{equation}
z_{\hat \varphi (\mathcal{F})}(G_1) = \varphi^{-1}(z_{\mathcal{F}}(G_2)).
\end{equation}
\end{lemma}

\begin{proof}
Since $\hat \varphi\colon \hat G_2 \rightarrow \hat G_1$ is defined by
$\hat \varphi(x)=x \circ \varphi$, 
\begin{align}
\chi\in z_{\hat \varphi (\mathcal{F})}(G_1) & \Longleftrightarrow 
\chi(\hat\varphi (\mathcal{F})) \longrightarrow 0 \Longleftrightarrow 
\varphi(\chi)(\mathcal{F})\longrightarrow 0\Longleftrightarrow 
\varphi(\chi) \in z_{\mathcal{F}}(G_2),
\end{align}
as desired.
\end{proof}


A {\em continuity type} $\mathcal{T}$ on a subcategory $\mathsf{A}$
of $\mathsf{Grp(Top)}$
is an assignment of a collection $\mathcal{T}(G)$ of filter-bases to 
each $G \in \mathsf{A}$ in a way that 
$\varphi(\mathcal{F})\in \mathcal{T} (G_2)$ for 
every morphism $\varphi\colon G_1 \rightarrow G_2$ in $\mathsf{A}$
and every $\mathcal{F}\in\mathcal{T}(G_1)$.  
Examples~\ref{example:S} and~\ref{example:C} 
below explain the motivation behind the term ``continuity type."





\begin{proposition} \label{prop:f-closure}
Let $\mathcal{T}$ be a continuity type on $\mathsf{Ab(Top)}$.
For every $G \in \mathsf{Ab(Top)}$ and $H \leq G$, put
\begin{equation}
\mathfrak{f}_G(H)=\bigcap\{z_\mathcal{F}(G)\mid\mathcal{F}\in\mathcal{T}(\hat G),
H \leq z_\mathcal{F}(G)\}.
\end{equation}
Then $\mathfrak{f}$ is an idempotent
closure operator on $\mathsf{Ab(Top)}$.
\end{proposition}

We call $\mathfrak{f}$ the closure operator 
{\em associated with $\mathcal{T}$}.

\begin{proof}
Clearly, $\mathfrak{f}$ is extensive and monotone, so we prove that
each continuous homomorphism $\varphi\colon G_1 \rightarrow G_2$ 
is $\mathfrak{f}$-continuous. Let $H \leq G_2$.
Since $\hat \varphi \colon \hat G_2 \rightarrow \hat G_1$ is continuous,
$\hat\varphi(\mathcal{F}_2)\in \mathcal{T}(\hat G_1)$ for every
$\mathcal{F}_2\in \mathcal{T}(\hat G_2)$. Thus, 
\begin{align}
\mathfrak f_{G_1}(\varphi^{-1}(H)) & = 
\bigcap\{z_{\mathcal{F}_1}(G_1)\mid\mathcal{F}_1\in\mathcal{T}(\hat G_1),
\varphi^{-1}(H) \leq z_{\mathcal{F}_1}(G_1)\} \\
&  \subseteq \bigcap\{z_{\hat\varphi(\mathcal{F}_2)}(G_1)\mid
\hat\varphi(\mathcal{F}_2) \in\mathcal{T}(\hat G_1),
\varphi^{-1}(H) \leq z_{\hat\varphi (\mathcal{F}_2)}(G_1)\} \\
& \stackrel  {\ref{lemma:fCLOP:funct}} =
\bigcap\{\varphi^{-1}(z_{\mathcal{F}_2}(G_2))\mid
\mathcal{F}_2 \in\mathcal{T}(\hat G_2),
(H) \leq z_{\mathcal{F}_2}(G_2)\} = 
\varphi^{-1} (\mathfrak f_{G_2}(H)),
\end{align}
as desired.  Obviously, $\mathfrak{f}$ is idempotent 
(cf. \cite[Exercise~2.D(a)]{DT}).
\end{proof}





A continuity type $\mathcal{T}$ is {\em regular on $G$} if 
$\mathcal{T}(G)=\mathcal{T}(G_d)$, where $G_d$ is the group $G$ with
discrete topology; a subgroup $N$ of $G \in \mathsf{Ab(Top)}$ is 
{\em $\mathcal{T}$-dually embedded} into $G$ if for every
$\mathcal{F}_2 \in \mathcal{T}(\hat N)$, there is 
$\mathcal{F}_1 \in \mathcal{T}(\hat G)$ such that
$\hat i_N(\mathcal{F}_1)=\mathcal{F}_2$. 



If $G \in \mathsf{Ab(Top)}$, then $G^+$ is the group $G$ with the 
topology of pointwise convergence on $\hat G$, and
$\mathbf{n}(G)=\bigcap\limits_{x \in \hat G} \ker x$.
It follows from the universal
property of $bG$ that $\widehat{bG}=\hat G_d$. 


\begin{proposition} \label{prop:contype}
Let $\mathcal{T}$ be a continuity type on $\mathsf{Ab(Top)}$, 
$\mathfrak{f}$ its associated closure operator, and
$G \in \mathsf{Ab(Top)}$.

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
One has $\mathbf{n}(G)\subseteq \mathfrak f_G(\{0\})$.

\item
If every principal ultrafilter on $\hat G$ is in $\mathcal{T}(\hat G)$,
then $\mathbf{n}(G)=\mathfrak f_G(\{0\})$.

\item
If \,$\mathcal{T}$ is regular on $\hat G$ , then 
$\mathfrak f_G(H) = \mathfrak f_{G^+}(H)$ for every $H\leq G$.

\item
If\, $\mathcal{T}$ is regular, then 
every dense subgroup of\, $G$ is $\mathcal{T}$-dually embedded
into $G$.

\item
If $N$ is $\mathcal{T}$-dually embedded into
$G$, then $\mathfrak{f}_N(H)=N\cap\mathfrak{f}_G(H)$ for
every $H\leq N$.

\end{list}
\end{proposition}


\begin{proof}
(a) If $\chi \in \mathbf{n}(G)$, then $\chi(\mathcal{F}) = \{ \{0\} \}$ for
every filter-base $\mathcal{F}$ on $\hat G$, and so $\chi\in z_\mathcal{F}(G)$. 
Thus, $\mathbf{n}(G)\subseteq z_\mathcal{F}(G)$ for every
$\mathcal{F}$, and therefore 
$\mathbf{n}(G)\subseteq \bigcap\limits_{\mathcal{F}\in\mathcal{T}(\hat G)}
z_\mathcal{F}(G)=\mathfrak f_G(\{0\})$.

(b) Let $x \in \hat G$. By the assumption, the principal ultrafilter
$\dot x \in \mathcal{T}(\hat G)$, and $z_{\dot {x}}(G) = \ker x$. 
Thus, $\mathfrak f_G(\{0\}) \subseteq \ker x$. This holds for 
every $x \in \hat G$, and therefore (using (a))
$\mathbf{n}(G)=\mathfrak f_G(\{0\})$.

(c) The continuous homomorphisms $G \rightarrow \hat G \rightarrow bG$ give 
rise to  $\hat G_d=\widehat{bG}\rightarrow\widehat{G^+}\rightarrow\hat G$, and 
thus $\mathcal{T}(\hat G_d)\subseteq\nolinebreak\mathcal{T}(\widehat{G^+})
\subseteq\mathcal{T}(\hat G)$. Since $\mathcal{T}$ is regular on 
$\hat G$, the two ends of this inclusion are equal, and therefore
$\mathcal{T}(\widehat{G^+})=\mathcal{T}(\hat G)$.
The definition of $z_\mathcal{F}$ is independent of the topology,
and thus $z_\mathcal{F}(G)=\nolinebreak z_\mathcal{F}(G^+)$ for every 
filter-base $\mathcal{F}$ on the \underline{set} $\hat G=\widehat{G^+}$. 
Hence, the collections whose intersection give 
$\mathfrak f_G(H)$ and $\mathfrak f_{G^+}(H)$ coincide.


(d) Let $D \leq G$ be a dense subgroup. Then
$\hat G$ and $\hat D$ have the same underlying group, and so
$\mathcal{T}(\hat D)=\mathcal{T}(\hat D_d)=\mathcal{T}(\hat G_d)=
\mathcal{T}(\hat G)$, because $\mathcal{T}$ is regular.


(e) Clearly, $\mathfrak{f}_N(H)\subseteq \mathfrak{f}_G(H)$, because
the inclusion $N \rightarrow G$ is $\mathfrak f$-continuous. Let
$\mathcal{F}_2 \in \mathcal{T}(\hat N)$, and let 
$\mathcal{F}_1\in\mathcal{T}(\hat G)$ be such that 
$\hat i_N(\mathcal{F}_1)=\mathcal{F}_2$ (the existence of
$\mathcal{F}_1$ is guaranteed as $N$ is $\mathcal{T}$-dually embedded
into $G$). By Lemma~\ref{lemma:fCLOP:funct},  $z_{\mathcal{F}_2}(N)=
z_{\hat i_{\empty_N} (\mathcal{F}_1)}(N)=i_N^{-1}(z_{\mathcal{F}_1}(G))=
N \cap z_{\mathcal{F}_1}(G)$, so if $H \leq z_{\mathcal{F}_2}(N)$,
then $z_{\mathcal{F}_2}(N)\supseteq N \cap \mathfrak{f}_G(H)$.
Therefore, $\mathfrak{f}_N(H)\supseteq  N\cap \mathfrak{f}_G(H)$,
as desired.
\end{proof}



\begin{theorem} \label{thm:reduce:bohr}
Let $\mathcal{T}$ be a regular continuity type on $\mathsf{Ab(Top)}$ and
$\mathfrak{f}$ its associated closure operator. Then
$\mathfrak{f}_G(H) = \rho^{-1}_G(\mathfrak{f}_{bG}(\rho_G(H)))$
for every abelian topological group $G$ and subgroup $H \leq G$.
\end{theorem}

\begin{proof}
Set $b^+G=\rho_G(G)$; it is a dense subgroup of $bG$. Observe that
$bG = b(G^+)$ (and so $b^+(G^+)=b^+ G$), and $b^+G^+=G^+/\mathbf{n}(G)$.
Since $b^+(G^+)$ is dense in $b(G^+)$,
by Proposition~\ref{prop:contype}(d), $b^+(G^+)$ is 
$\mathcal{T}$-dually embedded into $bG$. Thus, one has
\begin{align}
\mathfrak{f}_G(H)
\stackrel{\rm \ref{prop:contype}(c)} = 
\mathfrak{f}_{G^+}(H)
& \hspace{6pt} =\hspace{6pt}
\rho^{-1}_{G^+}(\mathfrak{f}_{b^+(G^+)}(\rho_{G^+}(H)))  
\label{eq:exp:1}\\
& \stackrel {\rm \ref{prop:contype}(e)}=
\rho^{-1}_G(\mathfrak{f}_{bG}(\rho_G(H))\cap b^+G) 
 = \rho^{-1}_G(\mathfrak{f}_{bG}(\rho_G(H))).
\end{align}
Note that the last equality in (\ref{eq:exp:1}) holds because
every continuous character of $G^+$ factors through 
$\rho_{G^+}$ (so $\widehat{G^+}=\widehat{b^+(G^+)}$ as sets), 
and $\mathcal{T}$ is regular.
\end{proof}


\begin{example}
For every topological group $G$, put $\mathcal{D}(G)$ to be the set of all
filter-bases on $G$. The associated closure operator is 
discrete on every $K \in \mathsf{Ab(CompH)}$, because for every 
(not necessary closed)
subgroup $H$, there is a filter $\mathcal{F}$ on $A=\hat K$
such that $H=z_\mathcal{F}(K)$ (cf.~\cite[2.1]{BeiSteWin}). 
Therefore, by Theorem~\ref{thm:reduce:bohr},
for $G \in \mathsf{Ab(Top)}$,
$\mathfrak{f}^\mathcal{D}_G(H)=\rho^{-1}_G(\rho_G(H))=H+\mathbf{n}(G)$. Hence,
$\mathfrak{f}^\mathcal{D}_G(H) = c_\mathbf{n}$, the minimal closure operator
associated with $\mathbf{n}$ (cf.~\cite[3.5-3.6]{DikCOTA}).
\end{example}


Let $\mathsf{A}$ be a subcategory of $\mathsf{Grp(Top)}$ and $\mathcal{T}$ be
a continuity type on $\mathsf{A}$. For $G_1,G_2 \in \mathsf{A}$,
we say that a (not necessarily continuous)
homomorphism $\varphi\colon G_1 \rightarrow G_2$ is 
{\em $\mathcal{T}$-continuous} if $\varphi(\mathcal{F})\longrightarrow e$ 
in $G_2$ for every $\mathcal{F}\in\mathcal{T}(G_1)$ 
such that $\mathcal{F}\longrightarrow e$ in $G_1$. 

\begin{example} \label{example:S}
For every topological group $G$, put $\mathcal{S}(G)$ to be the set of all
filter-bases on $G$ generated by sequences (i.e., of the form 
$\{\{a_n \mid n \geq m\}\mid m \in\mathbb{N}\}$, where $\{a_n\}\subseteq G$).
A homomorphism $\varphi\colon G_1 \rightarrow G_2$ is $\mathcal{S}$-continuous
if and only if it is  sequentially continuous.
For $G\in\mathsf{Ab(Top)}$ and $\underline u \in \hat G^\mathbb{N}$,
one has $s_{\underline u}(G)=z_\mathcal{F}(G)$, where 
$\mathcal{F}\in\mathcal{S}(G)$ is the filter base generated by the 
sequence $\underline u$, and vice versa, to each 
$\mathcal{F}\in\mathcal{S}(G)$ there is a corresponding sequence that generates
it. Therefore, the associated closure operator of $\mathcal{S}$ is 
precisely $\mathfrak{g}$, which was defined earlier.
\end{example}

A map $f\colon X \rightarrow Y$ between topological spaces is said to be
{\em countably continuous} if $f$ is continuous on every
countable subset of $X$.

\begin{example} \label{example:C}
For every topological group $G$, put $\mathcal{C}(G)$ to be the set
of filter-bases on $G$ that contain a countable subset of $G$. 
In Lemma~\ref{lemma:l:cc}, we show that a homomorphism 
$\varphi\colon G_1 \rightarrow G_2$ is $\mathcal{C}$-continuous 
if and only if it is countably continuous. Moreover, in 
Theorem~\ref{thm:char:l}, we 
prove that the closure operator $\mathfrak{l}$ associated with 
$\mathcal{C}$  coincides with the $G_\delta$-closure on precompact 
Hausdorff abelian groups.
\end{example}


\begin{convention}
Until the end of the section, $\mathcal{T}$ is 
a fixed regular continuity type on $\mathsf{Ab(Top)}$,
and $\mathfrak{f}$ is its associated closure operator.
\end{convention}



Motived by Theorem~\ref{thm:reduce:bohr},  we confine 
our attention to $\mathsf{Ab(CompH)}$.
In the sequel, we put $\mathcal{F} \stackrel H \longrightarrow a_0$
whenever a filter-base $\mathcal{F}$
converges to a point $a_0$ in $A$ in the topology $\tau_H$.

\begin{samepage}
\begin{lemma} \label{lemma:f}
Let $K \in \mathsf{Ab(CompH)}$, $A=\hat K$, and $H \leq K$. Put
$\mathcal{T}_H(A) = \{\mathcal{F}\in\mathcal{T}(A)\mid 
\mathcal{F} \stackrel H \longrightarrow 0 \}$. Then:

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
$\mathfrak f_K(H)=\{ \chi \in K\mid \chi(\mathcal{F}) \longrightarrow 0
\text{ for all } \mathcal{F}\in \mathcal{T}_H(A) \}$;

\item
$\mathfrak f_K(H)=\{ \chi \in K\mid \chi\colon(A,\tau_H) 
\rightarrow \mathbb{T} \text{ is $\mathcal{T}$-continuous}\}$;

\item
the identity homomorphism $(A,\tau_H)\rightarrow 
(A,\tau_{\mathfrak{f}_{\empty_K}(H)})$ is $\mathcal{T}$-continuous.

\end{list}
\end{lemma}
\end{samepage}

\begin{proof}
(a) For $\mathcal{F}\in \mathcal{T}(A)$, $H \leq z_\mathcal{F}(K)$ if and only if
$\chi(\mathcal{F}) \longrightarrow 0$ for every $\chi \in H$---in other words,
$\mathcal{F} \stackrel H \longrightarrow 0$, which is the same as
$\mathcal{F}\in\mathcal{T}_H(A)$. Therefore,
\begin{align}
\mathfrak{f}_K(H)=\bigcap\{ z_\mathcal{F}(K) \mid 
\mathcal{F}\in \mathcal{T}_H(A)\} =
\{\chi \in K \mid \chi(\mathcal{F})\longrightarrow 0 \text{ for all }
\mathcal{F} \in \mathcal{T}_H(A)\}.
\end{align}

(b) Let $\chi \in \mathfrak{f}_K(H)$, and suppose that 
$\mathcal{F} \in \mathcal{T}(A)$ is such that $\mathcal{F}\longrightarrow 0$
in $\tau_H$. Then $\mathcal{F}\in\nolinebreak\mathcal{T}_H(A)$, and so
$\chi(\mathcal{F})\longrightarrow 0$ by (a). Thus,
$\chi$ is $\mathcal{T}$-continuous. Conversely, 
suppose that $\chi\colon (A,\tau_H)\rightarrow \mathbb{T}$ is 
$\mathcal{T}$-continuous, and let $\mathcal{F}\in \mathcal{T}_H(A)$.
Then $\chi(\mathcal{F})\longrightarrow 0$, because $\chi$ is 
$\mathcal{T}$-continuous, and therefore $\chi\in \mathfrak{f}_K(H)$
by (a).

(c) If $\mathcal{F}\in \mathcal{T}_H(A)$, then by (a),
$\chi(\mathcal{F}) \longrightarrow 0$ for every 
$\chi \in \mathfrak{f}_K(H)$, and therefore
$\mathcal{F} \stackrel {\mathfrak{f}(H)}\longrightarrow 0$.
\end{proof}


The next theorem is an immediate consequence of Lemma~\ref{lemma:f}(b).

\begin{theorem} \label{thm:f}
Let $K \in \mathsf{Ab(CompH)}$ and $A=\hat K$.
A subgroup $H \leq K$ is $\mathfrak{f}$-closed in $K$ if and only if every 
$\mathcal{T}$-continuous character of $(A,\tau_H)$ is continuous.
\qed
\end{theorem}



\begin{proposition} \label{prop:f-funct}
Let $A$ and $B$ be discrete abelian groups, $K=\hat A$ 
and $C=\hat B$ their Pontryagin duals, $H \leq K$, and $L\leq C$. For every
homomorphism $\varphi\colon A \rightarrow B$,
the following statements are equivalent:

\begin{list}{{\rm (\roman{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item $\varphi\colon (A,\tau_H) \rightarrow (B,\tau_L)$ is 
$\mathcal{T}$-continuous;

\item $\chi\circ\varphi$ is $\mathcal{T}$-continuous on $(A,\tau_H)$ for
every $\mathcal{T}$-continuous character $\chi$ of $(B,\tau_L)$;

\item
$\mathfrak{f}\varphi\colon(A,\tau_{\mathfrak{f}_{\empty_K}(H)})\rightarrow 
(B,\tau_{\mathfrak{f}_{\empty_C}(L)})$ is continuous.
\end{list}
\end{proposition}


\begin{proof}
Clearly, (i) $\Rightarrow$ (ii), because the composite of two
$\mathcal{T}$-continuous homomorphisms is $\mathcal{T}$-continuous. 
The implication (ii) $\Rightarrow$ 
(iii) follows from Lemma~\ref{lemma:f}(b), because (ii) states that
$\chi\circ\varphi \in \mathfrak{f}_K(H)$ for every
$\chi \in \mathfrak{f}_C(L)$. To show (iii) $\Rightarrow$ (i), notice that
$\varphi$ is the composite of\linebreak 
$(A,\tau_H)\rightarrow (A,\tau_{\mathfrak{f}_{\empty_K}(H)})$, 
$\mathfrak f \varphi$, 
and $(B,\tau_{\mathfrak{f}_{\empty_C}(L)}) \rightarrow (B,\tau_L)$. Since these  
three maps are all $\mathcal{T}$-continuous (cf. Lemma~\ref{lemma:f}(c)), 
their composite is also $\mathcal{T}$-continuous.
\end{proof}

Inspired by the notions of $k$-group and $s$-group introduced by Noble 
\cite{Noble3} \& \cite{Noble}, and the term ``$kk$-group" used by Deaconu 
\cite{PHDDeaconu},
we say that a topological group $G$ is a {\em $\mathcal{T}k$-group} 
if every $\mathcal{T}$-continuous homomorphism of $G$ into a 
compact Hausdorff group is continuous.

\pagebreak[2]

\begin{proposition} \label{prop:Tk-grp}
Let $K \in \mathsf{Ab(CompH)}$, $A=\hat K$, and $H \leq K$.
The following are equivalent:

\begin{list}{{\rm (\roman{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
every $\mathcal{T}$-continuous homomorphism 
$\varphi\colon (A,\tau_H) \rightarrow (B,\tau_L)$ into a precompact 
abelian group (where $B$ is a discrete group and $L \leq \hat B$) 
is continuous;

\item
$(A,\tau_H)$ is an $\mathcal{T}k$-group;


\item
every $\mathcal{T}$-continuous character
$\chi\colon (A,\tau_H) \rightarrow \mathbb{T}$ is continuous.

\end{list}
\end{proposition}





\begin{proof}
(i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) is obvious, because
every compact group is precompact, and $\mathbb{T}$ is compact.
In order to show (iii) $\Rightarrow$ (i), let
$\varphi\colon (A,\tau_H) \rightarrow (B,\tau_L)$  be an 
$\mathcal{T}$-continuous
homomorphism. By Proposition~\ref{prop:f-funct}, 
$\chi \circ \varphi$ is $\mathcal{T}$-continuous on $(A,\tau_H)$ for 
every character $\chi \in L$ of $(B,\tau_L)$. Applying (iii) to 
the character $\chi \circ \varphi$ yields that it is continuous, in other
words, $\chi \circ \varphi \in H$. Therefore, $\chi\circ\varphi \in H$ 
for every $\chi \in L$, and hence $\varphi$ is continuous.
\end{proof}




A combination of Theorem~\ref{thm:f} and Proposition~\ref{prop:Tk-grp} 
yields:

\begin{theorem} \label{thm:Tk-closed}
Let $K \in \mathsf{Ab(CompH)}$ and $A=\hat K$.
A subgroup $H\leq K$ is $\mathfrak{f}$-closed if and only if\,
$(A,\tau_H)$ is an $\mathcal{T}k$-group. In particular,
$(A,\tau_{\mathfrak f_{\empty_K}(H)})$ is an $\mathcal{T}k$-group for every
$H \leq K$.
\qed
\end{theorem}



\bigskip




Theorem~\ref{thm:subcat} below is modeled on the relationship between the 
category of Hausdorff topological spaces with $k$-continuous maps,
Hausdorff topological spaces with continuous maps, and Hausdorff $k$-spaces
(cf.~\cite[7.2]{Bor2}).
In what follows, $\mathsf{Pr_\mathcal{T}Ab}$ is the category of precompact abelian 
groups and their $\mathcal{T}$-continuous homomorphisms, $\mathsf{PrAb}$ stands 
for the category of precompact abelian groups and their continuous homomorphisms,
and $\mathsf{\mathcal{T}kPrAb}$ is the full
subcategory of $\mathsf{PrAb}$ consisting of the $\mathcal{T}k$-groups. 




\begin{theorem} \label{thm:subcat}
The assignment
\begin{align*}
\mathfrak f \colon \mathsf{Pr_\mathcal{T}Ab} & 
\longrightarrow \mathsf{\mathcal{T}kPrAb} \\
\hspace{120pt}
(A,\tau_H) & \longmapsto (A,\tau_{\mathfrak f_{\empty_K} (H)}), 
\quad (K=\hat A,H\leq K)
\end{align*}
is a functor. Furthermore,

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item 
$\mathfrak f \colon \mathsf{Pr_\mathcal{T}Ab}  \longrightarrow 
\mathsf{\mathcal{T}kPrAb}$ is  an equivalence of categories;

\item
$\mathfrak f \colon \mathsf{PrAb}  \longrightarrow 
\mathsf{\mathcal{T}kPrAb}$ is a coreflection.
\end{list}
\end{theorem}




\begin{proof}
It was explained earlier that every precompact abelian group has the form
$(A,\tau_H)$, where $A$ is a discrete abelian group and  $H \leq \hat A$
($A$ and $H$ are uniquely determined). By Theorem~\ref{thm:Tk-closed},
$\mathfrak{f} (A,\tau_H) = (A,\tau_{\mathfrak{f}_{\empty_K}(H)}) 
\in \mathsf{\mathcal{T}kPrAb}$,
and by Proposition~\ref{prop:f-funct}, $\mathfrak{f}\varphi$ is continuous 
for every $\mathcal{T}$-continuous homomorphism 
$\varphi\colon (A,\tau_H) \rightarrow (B,\tau_L)$.
Therefore, $\mathfrak{f}$ is well-defined, and it is a functor.

(a) Clearly, $\mathfrak{f}$ and the inclusion 
$I\colon \mathsf{\mathcal{T}kPrAb} \longrightarrow \mathsf{Pr_\mathcal{T}Ab}$ are
faithful, and by Proposition~\ref{prop:f-funct}, they are also full. Since 
$\mathfrak{f}$ is idempotent, 
$\mathfrak{f}I=\operatorname{Id}_{\mathsf{\mathcal{T}kPrAb}}$.
By Lemma~\ref{lemma:f}(c), $(A,\tau_H)$ and 
$(A,\tau_{\mathfrak{f}_{\empty_K}(H)})$
are isomorphic in $\mathsf{Pr_\mathcal{T}Ab}$, and therefore
$I \mathfrak{f} \cong \operatorname{Id}_{\mathsf{Pr_\mathcal{T}Ab}}$. Hence,
$\mathfrak{f}$ is an equivalence of categories.

(b) For $(B,\tau_L)\in \mathsf{PrAb}$ (where $B$ is a discrete abelian group
and $L \leq C:=\hat B$),
the counit $\gamma_{(B,\tau_L)}\colon (B,\tau_{\mathfrak f_{\empty_C}(L)}) 
\rightarrow (B,\tau_L)$ is continuous (because $L \leq \mathfrak f_C(L)$).
By Proposition~\ref{prop:f-funct}, every continuous
homomorphism $\varphi\colon (A,\tau_H) \rightarrow (B,\tau_L)$ 
gives rise to a continuous homomorphism 
$\mathfrak f \varphi \colon (A,\tau_{\mathfrak{f}_{\empty_K}(H)}) 
\rightarrow (B,\tau_{\mathfrak f_{\empty_C}(L)})$. Whenever $(A,\tau_H)$ is 
an $\mathcal{T}k$-group, $\mathfrak{f}_K(H)=H$ (cf. Theorem~\ref{thm:Tk-closed}), 
and therefore $\varphi$ factors through 
$\gamma_{(B,\tau_L)}$ uniquely.
\end{proof}




%01234567890123456789012345678901234567890123456789012345678901234567890123456789



\section{Application I: The $\mathbf{G_\delta}$-closure}

\label{sect:realcompact}

Recall that for a topological space $X$, the {\em $G_\delta$-closure} 
$\bar A^\delta$ of $A \subseteq X$ consists of all  points $x \in X$ such 
that each $G_\delta$-set that contains $x$ meets $A$; $A$ is {\em
$G_\delta$-dense} in $X$ if $\bar A ^\delta =X$. 


Let $\mathcal{C}$ be the continuity type described in 
Example~\ref{example:C}, and set $\mathfrak{l}$ to be 
its associated closure operator. In this section, we study the 
closure operator $\mathfrak l$ on precompact Hausdorff abelian groups.


\begin{theorem} \label{thm:char:l}
Let $P$ be a precompact Hausdorff abelian group. Then
$\mathfrak l_P(H) = \bar H^\delta$ for every subgroup $H \leq P$. In other 
words, $\mathfrak l$ coincides with the $G_\delta$-closure on precompact 
Hausdorff abelian groups.
\end{theorem}




Before proving Theorem~\ref{thm:char:l}, we present some basic properties 
of $\mathfrak{l}$.
Recall that a subgroup $N$ of $G\in\mathsf{Ab(Top)}$ is said to be
{\em dually embedded} if $\hat i_N\colon \hat G \rightarrow \hat N$
is surjective, where $i_N$ is the inclusion. 


\begin{proposition} \label{prop:l:basic} 
For every $G\in\mathsf{Ab(Top)}$:

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
$\mathfrak{l}_G(\{0\})=\mathbf{n}(G)$;

\item
$\mathfrak{l}_G(H)=\mathfrak{l}_{G^+}(H)$ for every $H\leq G$;

\item
if $N\leq G$ is dually embedded in $G$, then 
$\mathfrak{l}_N(H)=\mathfrak{l}_G(H)\cap N$ for every $H \leq N$.

\end{list}
\end{proposition}


\begin{proof}
(a) and (b) follow from Proposition~\ref{prop:contype} (b) and (c), respectively.
By Proposition~\ref{prop:contype}(e), in order to show (c), it
suffices to show that $N$ is $\mathcal{C}$-dually embedded in $G$.
To that end, let $\mathcal{F}_2 \in \mathcal{C}(\hat N)$,
and let $D_2 \in \mathcal{F}_2$ be a countable subset.
Since $\hat i_N\colon \hat G \rightarrow \hat N$ is surjective, 
there exists a countable subset $D_1$ of $\hat G$ such that 
$\hat i(D_1)=D_2$. It is easy to see that for 
\begin{align}
\mathcal{F}_1=\{\hat i_N^{-1}(F) \mid F \in \mathcal{F}_2\} \cup
\{\hat i_N^{-1}(F)\cap D_1 \mid F \in \mathcal{F}_2\},
\end{align}
$\mathcal{F}_1 \in \mathcal{C}(\hat G)$, and 
$\hat i_N(\mathcal{F}_1)=\mathcal{F}_2$.
\end{proof}


It follows by Pontryagin duality that
every subgroup of a compact Hausdorff abelian group
is dually embedded, and thus Proposition~\ref{prop:l:basic}(c)
yields:

\begin{corollary} \label{cor:l:her}
For every $K\in \mathsf{Ab(CompH)}$ and subgroups
$H\leq N \leq K$,
$\mathfrak{l}_N(H)=\mathfrak{l}_K(H)\cap N$ holds.
In other words, $\mathfrak{l}$ is hereditary on
$\mathsf{Ab(CompH)}$, and thus $\mathfrak{l}$ is hereditary on every 
precompact Hausdorff abelian group.
\qed
\end{corollary}




\begin{lemma} \label{lemma:l:cc}
A homomorphism $\varphi\colon G_1 \rightarrow G_2$ 
is $\mathcal{C}$-continuous if and only if it is 
countably continuous.
\end{lemma}

\begin{proof}
Let $D$ be a countable subset of $G_1$, and set $S=\langle D \rangle$,
the (countable) subgroup generated by $D$. Every filter-base
$\mathcal{F}$ on $S$ belongs to $\mathcal{C}(G_1)$. Thus,  if
$\mathcal{F}\longrightarrow e$ in $S$, then
$\varphi(\mathcal{F}) \longrightarrow e$ in $G_2$, because
$\varphi$ is $\mathcal{T}$-continuous. Therefore, $\varphi\rest_S$ is 
continuous, and in particular $\varphi\rest_D$ is continuous.

Conversely, suppose that $\varphi$ is countably continuous, and
let $\mathcal{F}\in \mathcal{C}(G_1)$ such that 
$\mathcal{F} \longrightarrow e$ in $G_1$. Let $D\in \mathcal{F}$ be countable.
By replacing $\mathcal{F}$ with the filter-base 
$\{F \in \mathcal{F} \mid F \subseteq D\}$, we may assume that every set in
$\mathcal{F}$ is contained in $D$. Set $D_0=D\cup\{e\}$. Since 
$\varphi$ is countably continuous,  $\varphi\rest_{D_0}$ is continuous. 
Thus, $\varphi\rest_{D_0}(\mathcal{F})\longrightarrow e$, because
$\mathcal{F}\longrightarrow e$ in $D_0$. 
\end{proof}




\begin{proof}[Proof of Theorem~\ref{thm:char:l}.] %{\em Step 1.}
Since both $\mathfrak l$ and the $G_\delta$-closure are hereditary on 
precompact Hausdorff groups (cf. Corollary~\ref{cor:l:her}), it suffices 
to show the theorem for the case where $P=K$ is compact. 
Set $A=\hat K$. By Lemma~\ref{lemma:f}(b), $\chi \in \mathfrak l_K(H)$ if 
and only if $\chi$ is $\mathcal C$-continuous with respect to $\tau_H$. By 
Lemma~\ref{lemma:l:cc}, this amounts to $\chi$ being countably continuous 
on $(A,\tau_H)$. We follow the idea of \cite[3.1]{ComfHernTrig} in order 
to show that the last statement is equivalent to $\chi \in \bar H^\delta$.

%{\em Step 2.}
Since $K \in \mathsf{Ab(CompH)}$, $K$ embeds into the product
$\mathbb{T}^A$. For each $(t_a)_{a\in A} \in \mathbb{T}^A$ and countable 
subset $C \subseteq A$, the set
$\{(s_a)_{a\in A}\mid s_c=t_c \mbox{ for all } c \in C \}$
is $G_\delta$ in $\mathbb{T}^A$ (because $\mathbb{T}$ has a countable 
pseudocharacter), and these sets form a base for the $G_\delta$-topology 
on $\mathbb{T}^A$. Thus, for every countable subset $C \subseteq A$,
\begin{equation}
N(\chi,C)=\{\psi \in K \mid \psi\rest_C = \chi\rest_C\}
\end{equation}
is a basic $G_\delta$-neighborhood of $\chi$ in $K$, and these sets form a 
base for the $G_\delta$-topology on $K$.

If $\chi \in \bar H^\delta$, then $N(\chi,C)\cap H \neq \emptyset$ for 
every countable subset $C$ of $A$, and therefore for every $C$ there is
$\psi_C \in H$ such that $\psi_C \rest_C=\chi \rest_C$. In particular,
$\chi\rest_C$ is continuous with respect to the induced 
topology $\tau_H\rest_C$, and hence $\chi$ is countably continuous.

Conversely, suppose that $\chi$ is countably continuous, and let 
$C\subseteq A$ be countable. By replacing $C$ with the subgroup that it 
generates, we may assume that $C$ is a countable subgroup of $A$, because
$N(\chi,\langle C \rangle)= N(\chi,C)$.
One has $\chi\rest_C \in H \rest_C$, because
$\chi\rest_C$ is continuous with respect to $\tau_H\rest_C$
and $(C, \tau_H\rest_C)$ is a precompact.
Therefore,  $H \cap N(\chi,C) \neq \emptyset$, as desired. 
\end{proof}




We proceed with presenting a characterization of $\mathfrak l$-closed and 
$\mathfrak l$-dense
subgroups of compact Hausdorff abelian groups. To that
end, we first formulate explicitly an argument that was used by Comfort,
Hern\'andez and Trigos-Arrieta \cite{ComfHernTrig}.  In order to do so, 
the following terminology is 
needed: For a topological space $X$, a {\em zero-set in $X$} is a set of 
the form $Z(f)=f^{-1}(0)$, where $f$ is a real-valued continuous function 
on $X$. A subset  $Y \subseteq X$ is {\em $z$-embedded} in $X$ if for 
every zero-set $Z$ in $Y$ there is a zero-set $W$ in $X$ such that 
$Z=W\cap Y$; $Y$ is {\em $C$-embedded} in $X$ if every continuous 
real-valued function on $Y$ extends continuously to $X$.
One says that $X$ is an {\em $Oz$-space} if every open 
subset of $X$ is $z$-embedded. 



\begin{proposition} \label{prop:HW-Gdelta}
Let $G$ be a Hausdorff group whose (Ra\u{\i}kov-)completion 
$\widetilde G$ is locally compact (in other words, $G$ is locally 
precompact), 
and let $H$ be its subgroup. Then: 

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
$H$ is $z$-embedded in $\tilde G$;

\item
$H$ is $C$-embedded in $\bar H^\delta$; 

\item 
$\upsilon H \subseteq \upsilon G$ and
$\bar H^\delta= \upsilon H \cap G$, where
$\upsilon$ stands for the Hewitt-realcompactification;

\item
if $G$ is realcompact, then $\bar H^\delta=\upsilon H$.

\end{list}

\end{proposition}

Note that in contrast to the rest of this paper, in this proposition, the
group $G$ is not assumed to be abelian. We claim no novelty for the proof
of Proposition~\ref{prop:HW-Gdelta} (which heavily relies on results
of Blair and Hager), and it is included here for the sake of
completeness.

\begin{flemma}[{\cite[1.1(a)-(b)]{BlaHag2}, \cite[2.1, 5.1]{Blair}}]

\label{flemma:BlaHag}

\mbox{ }

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}


\item
Every $z$-embedded $G_\delta$-dense subset of a topological space $X$ is 
$C$-embedded in $X$.

\item
If $Y$ is $z$-embedded in $X$, then $\upsilon Y \subseteq \upsilon X$
and $\upsilon Y$ is the $G_\delta$-closure of $Y$ in $\upsilon X$.

\item
A topological space $X$ is normal if and only if every closed subset of 
$X$ is $z$-embedded.

\item
A topological space $X$ is an $Oz$-space if and only if every dense subset 
of $X$ is $z$-embedded.


\end{list}

\end{flemma}




\begin{proof}
(a) The completion $\widetilde G$ is locally compact, 
and thus normal. By Lemma~\ref{flemma:BlaHag}(c), the closed
subgroup $\widetilde H$ of $\widetilde G$ is $z$-embedded in 
$\widetilde G$. By a result of Ross and Stromberg \cite{RossStro},
$\widetilde H$ is an $Oz$-space, because it is locally compact (as a 
closed subgroup of $\widetilde G$). Thus, the dense subgroup $H$ is 
$z$-embedded in $\tilde H$ (cf.  Lemma~\ref{flemma:BlaHag}(d)). Therefore,
$H$ is $z$-embedded in $\tilde G$.

(b) It follows from (a) that $H$ is $z$-embedded in $\bar H^\delta$, and 
therefore it is $C$-embedded in it, by  Lemma~\ref{flemma:BlaHag}(a).

(c) It follows from (a) that $H$ is $z$-embedded in $G$, and therefore the 
statement follows from Lemma~\ref{flemma:BlaHag}(b).


(d) follows from (c).
\end{proof}


\begin{corollary} \label{cor:l:misc}
Let $K\in \mathsf{Ab(CompH)}$ and $H\leq K$.

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
If $H$ is closed, then it is $\mathfrak l$-closed in $K$.

\item
$H$ is $\mathfrak l$-closed in $K$ if and only if $H$ is realcompact.

\item
$H$ is $\mathfrak l$-dense in $K$ if and only if it is dense in $K$ and 
pseudocompact.

\item
$K$ is metrizable if and only if every subgroup of $K$ is 
$\mathfrak l$-closed.

\end{list}

\end{corollary}





\begin{proof}
In light of Theorem~\ref{thm:char:l}, (a) is obvious, 
(b) follows from Proposition~\ref{prop:HW-Gdelta}(d), and 
(c) is a reformulation of a famous theorem of Comfort and Ross
\cite[1.2]{ComfRoss2}.

(d) If $K$ is metrizable, then every subgroup of $K$ is realcompact,
and thus, by (c), $\mathfrak{l}$-closed. Conversely, if every subgroup of 
$K$ is $\mathfrak{l}$-closed, then $K$ is hereditarily realcompact (by 
(b)), and therefore metrizable (cf.~\cite[3.3]{HernMaca}).
\end{proof}


We conclude this section by observing that a result of Hern\'andez and
Macario can be obtained as an easy consequence of the foregoing
discussion:

\begin{fcorollary}[{\cite[3.2]{HernMaca}}] \label{fcor:real:ccc}
Let $K\in\mathsf{Ab(CompH)}$ and set $A=\hat K$.
A subgroup $H \leq K$ is realcompact if and only if every countably 
continuous character of $(A,\tau_H)$ is continuous.
\end{fcorollary}

\begin{proof}
As we explained in the proof of Theorem~\ref{thm:char:l}, $\mathfrak
l_K(H)$ is the set of countably continuous characters of $(A,\tau_H)$ (cf.  
Lemma~\ref{lemma:f}(b) and Lemma~\ref{lemma:l:cc}). On the other hand, by
Corollary~\ref{cor:l:misc}(b), $H$ is realcompact if and only if it is
$\mathfrak l$-closed in $K$.
\end{proof}



%01234567890123456789012345678901234567890123456789012345678901234567890123456789



\section{Application II: The $\MakeLowercase{\mathfrak{g}}$-closure}

\label{sect:gclosure}


In this section, we apply the results of section~\ref{sect:cont}
to $\mathfrak{g}$, and  prove 
Theorems~\ref{thm:positive}-\ref{thm:gc:MAP}.
As shown in Example~\ref{example:S}, $\mathfrak{g}$ is the
closure operator associated with the continuity type $\mathcal{S}$
(assigning to a group the filter-bases generated by sequences).
Furthermore, $\mathcal{S}$-continuous homomorphisms are precisely
the sequentially continuous ones. 

\begin{theorem} \label{thm:g}
Let $K \in \mathsf{Ab(CompH)}$ and $A=\hat K$.
A subgroup $H \leq K$ is:

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item

$\mathfrak{g}$-closed in $K$ if and only if every 
sequentially continuous character of $(A,\tau_H)$ is continuous;

\item
$\mathfrak{g}$-dense in $K$ if and only if $(A,\tau_H)$ has no
non-trivial convergent sequences.

\end{list}
\end{theorem}

Note that part (b) of Theorem~\ref{thm:f} appeared earlier
in \cite[5.11]{BarDikMilWeb2}.

\begin{proof}
(a) is Theorem~\ref{thm:f} applied to $\mathfrak{g}$.

(b) If $(A,\tau_H)$ has no non-trivial convergent sequences, then
all its characters are sequentially continuous, 
and thus, by Lemma~\ref{lemma:f}(b), $\mathfrak g_K(H)=K$.
Conversely, suppose that $H$ is $\mathfrak{g}$-dense in $K$---in 
other words, $\mathfrak g_K(H)=K$. Then,
by Lemma~\ref{lemma:f}(c), the identity map
$(A,\tau_H)\rightarrow (A,\tau_{K})$ is sequentially continuous.
Thus, if $\{a_n\}$ is a convergent sequence with respect to
$\tau_H$, then it must converge in $\tau_K$. On the other hand,
one has $(A,\tau_K)=A^+ \subseteq bA$, and it is well known that 
$A^+$ has no convergent sequences, because $A$ is discrete abelian
(cf. \cite{Flor} and \cite{Glick}).
\end{proof}


It follows from the definition of $\mathcal{S}$ 
that it is regular, and every
dually embedded subgroup $N$ of a group $G\in\mathsf{Ab(Top)}$
is $\mathcal{S}$-dually embedded (because if
$\{u_n\}$ is a sequence in $\hat N$,
then there is a sequence $\{v_n\}$ in $\hat G$ such that
$\hat i_N(v_n)=u_n$). Therefore, the following well-known and basic properties of 
$\mathfrak{g}$ are immediate consequences of Proposition~\ref{prop:contype}
(cf.~\cite[2.9,2.11]{DikMilTon}):


\begin{proposition} \label{prop:gcl:basic} For every $G\in\mathsf{Ab(Top)}$:

\begin{list}{{\rm (\alph{enumi})}}
{\usecounter{enumi}\setlength{\labelwidth}{25pt}\setlength{\topsep}{-10pt}
\setlength{\itemsep}{-4pt} \setlength{\leftmargin}{25pt}}

\item
$\mathfrak{g}_G(\{0\})=\mathbf{n}(G)$;

\item
$\mathfrak{g}_G(H)=\mathfrak{g}_{G^+}(H)$ for every $H\leq G$;

\item
if $N\leq G$ is dually embedded in $G$, then 
$\mathfrak{g}_N(H)=\mathfrak{g}_G(H)\cap N$ for every $H \leq N$.
%---in 
%other words, $\mathfrak{g}$ is hereditary on $N$.
\qed
\end{list}
\end{proposition}


Now we are ready to prove the theorems that were formulated in
section~\ref{sect:intro}:


\begin{proof}[Proof of Theorem~\ref{thm:positive}.]
Let $H=\{h_1,h_2,\ldots\}$ be countable.  Since $K$
is compact, it follows from the Pontryagin
duality that its subgroups are dually embedded, and so
$\mathfrak g_{\bar H}(H)=\mathfrak g_K(H) \cap \bar H$ 
(by Proposition~\ref{prop:gcl:basic}). Every closed subgroup
of $K$ is $\mathfrak{g}$-closed (cf. \cite[2.12]{DikMilTon}),
and thus one has $g_K(H) \leq \bar H$. Therefore,
$\mathfrak g_{\bar H}(H)=\mathfrak g_K(H)$, and
without loss of generality, we may assume that $H$ is dense in $K$. 
In particular,  $(A,\tau_H)$ is Hausdorff.  
The topology $\tau_H$ is the initial one with respect to the homomorphism 
$(h_i)\colon A \rightarrow\mathbb{T}^\omega$. 
Thus, $(A,\tau_H)$ is metrizable, and in particular, every
sequentially continuous map on $(A,\tau_H)$ is continuous.
Therefore, by Theorem~\ref{thm:g}(a), $H$ is $\mathfrak{g}$-closed.
\end{proof}




\begin{proof}[Proof of Theorem~\ref{thm:gc-realcompact}.]
Let $H$ be a $\mathfrak g$-closed subgroup of $K$.
Since  $\mathcal{S}(\hat K) \subseteq \mathcal{C}(\hat K)$
(where $\mathcal{S}$ is as in Example~\ref{example:S}), one has
$H\leq \mathfrak l_K(H) \leq \mathfrak g_K(H) =H$, and thus
$H$ is $\mathfrak l$-closed in $K$. Therefore, by 
Corollary~\ref{cor:l:misc}(b), $H$ is realcompact. 
\end{proof}



\begin{proof}[Proof of Theorem~\ref{thm:gc:MAP}.]
(i) $\Rightarrow$ (ii): 
Let $H$ be a countable subgroup of $G$. 
By Theorem~\ref{thm:positive},  
$\rho_G(H)$ is $\mathfrak{g}$-closed in the Bohr-compactification $bG$, 
because it is countable---in other words,  
$\mathfrak g_{bG}(\rho_G(H))=\rho_G(H)$. Since
$G$ is maximally almost periodic (i.e., $\rho_G$ is injective),
one has $\rho_G^{-1}(\mathfrak g_{bG} (\rho_G(H)))=H$. Hence,
by Theorem~\ref{thm:reduce:bohr}, $H$ is $\mathfrak{g}$-closed
in $G$.


%Let $H$ be a countable subgroup of $G$. Since $G$ is maximally almost 
%periodic, $G^+$ is a Hausdorff precompact group. Let $K$ be the 
%completion of $G^+$; $H$ is a countable subgroup of the 
%a compact Hausdorff abelian group $K$, and thus,
%by Theorem~\ref{thm:positive},
%$H$ is $\mathfrak{g}$-closed in $K$. Therefore,
%$H$ is $\mathfrak{g}$-close in $G^+$,
%because $G^+$ is a subgroup of $K$. By
%Proposition~\ref{prop:gcl:basic}(b),
%$\mathfrak g_G(H)=\mathfrak g_{G^+}(H)$, and 
%hence $H$ is $\mathfrak{g}$-closed in $G$.




(ii) $\Rightarrow$ (iii) is trivial.


(iii) $\Rightarrow$ (i):
By (iii), the (cyclic) subgroup $\{0\}$ is $\mathfrak{g}$-closed
in $G$. Therefore, by Proposition~\ref{prop:gcl:basic}(a),
$\mathbf{n}(G)=\mathfrak g_G(\{0\})=\{0\}$, as desired.
\end{proof}









\section*{Acknowledgments}

I am deeply indebted to Professor Dikran Dikranjan, who introduced me 
to this topic and spent so much time corresponding with me about it.
Without his attention and encouragement, I would have struggled to 
carry out this research.

I wish to thank Professor Horst Herrlich, Professor Salvador
Hern\'andez and Dr. Gavin Seal for the valuable discussions 
and helpful suggestions that were
of great assistance in writing this paper.  


\bibliography{notes,notes2,notes3}



\begin{samepage}

{\bigskip\bigskip\noindent 
Department of Mathematics and Statistics\\
Dalhousie University\\
Halifax, B3H 3J5, Nova Scotia\\
Canada


\nopagebreak
\bigskip\noindent{\em e-mail: lukacs@mathstat.dal.ca} }
\end{samepage}



\end{document}


gclose.bbl
\def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{BarDikMilWeb}
Giuseppina Barbieri, Dikran Dikranjan, Chiara Milan, and Hans Weber.
\newblock Answer to {R}aczkowski's questions on convergent sequences of
  integers.
\newblock {\em Topology Appl.}, 132(1):89--101, 2003.

\bibitem{BarDikMilWeb2}
Giuseppina Barbieri, Dikran Dikranjan, Chiara Milan, and Hans Weber.
\newblock Convergent sequences in precompact group topologies.
\newblock Preprint.

\bibitem{BeiSteWin}
Mathias Beiglb\"ock, Christian Steineder, and Reinhard Winkler.
\newblock Sequences and filters of characters characterizing subgroups of
  compact abelian groups.
\newblock {\em Preprint}, ArXiv: math.GN/0412447.

\bibitem{BirDesSos}
A.~B{\'{\i}}r{\'o}, J.-M. Deshouillers, and V.~T. S{\'o}s.
\newblock Good approximation and characterization of subgroups of
  {$\mathbb{R}/\mathbb{Z}$}.
\newblock {\em Studia Sci. Math. Hungar.}, 38:97--113, 2001.

\bibitem{Blair}
Robert~L. Blair.
\newblock Spaces in which special sets are {$z$}-embedded.
\newblock {\em Canad. J. Math.}, 28(4):673--690, 1976.

\bibitem{BlaHag2}
Robert~L. Blair and Anthony~W. Hager.
\newblock Notes on the {H}ewitt realcompactification of a product.
\newblock {\em General Topology and Appl.}, 5:1--8, 1975.

\bibitem{Bor2}
Francis Borceux.
\newblock {\em Handbook of categorical algebra. 2}.
\newblock Cambridge University Press, Cambridge, 1994.
\newblock Categories and structures.

\bibitem{ComfHernTrig}
W.~W. Comfort, Salvador Hern{\'a}ndez, and F.~Javier Trigos-Arrieta.
\newblock Relating a locally compact abelian group to its {B}ohr
  compactification.
\newblock {\em Adv. Math.}, 120(2):322--344, 1996.

\bibitem{ComfRem}
W.~W. Comfort and Dieter Remus.
\newblock Compact groups of {U}lam-measurable cardinality: partial converses to
  theorems of {A}rhangel\cprime ski\u\i\ and {V}aropoulos.
\newblock {\em Math. Japon.}, 39(2):203--210, 1994.

\bibitem{ComfRoss}
W.~W. Comfort and K.~A. Ross.
\newblock Topologies induced by groups of characters.
\newblock {\em Fund. Math.}, 55:283--291, 1964.

\bibitem{ComfRoss2}
W.~W. Comfort and Kenneth~A. Ross.
\newblock Pseudocompactness and uniform continuity in topological a. groups.
\newblock {\em Pacific J. Math.}, 16:483--496, 1966.

\bibitem{PHDDeaconu}
Daniel Deaconu.
\newblock {\em On {LCA} groups and epimorphisms of topological groups}.
\newblock PhD thesis, York University, 2004.

\bibitem{DT}
D.~Dikranjan and W.~Tholen.
\newblock {\em Categorical structure of closure operators}, volume 346 of {\em
  Mathematics and its Applications}.
\newblock Kluwer Academic Publishers Group, Dordrecht, 1995.
\newblock With applications to topology, algebra and discrete mathematics.

\bibitem{Diktoptor}
Dikran Dikranjan.
\newblock Topologically torsion elements of topological groups.
\newblock In {\em Proceedings of the 16th Summer Conference on General Topology
  and its Applications (New York)}, volume~26, pages 505--532, 2001/02.

\bibitem{DikCOTA}
Dikran Dikranjan.
\newblock Closure operators in topological groups related to von neumann's
  kernel.
\newblock {\em Topology Appl.}, (to appear).

\bibitem{DikSan}
Dikran Dikranjan and Raffaele Di~Santo.
\newblock A characterization of the circle group via uniqueness of roots.
\newblock Preprint.

\bibitem{DikKun}
Dikran Dikranjan and Kenneth Kunen.
\newblock Characterizing subgroups of compact abelian groups.
\newblock {\em Preprint}, ArXiv: math.GN/0412327.

\bibitem{DikMilTon}
Dikran Dikranjan, Chiara Milan, and Alberto Tonolo.
\newblock A characterization of the maximally almost periodic abelian groups.
\newblock {\em J. Pure Appl. Algebra}, 197(1-3):23--41, 2005.

\bibitem{Flor}
Peter Flor.
\newblock Zur {B}ohr-{K}onvergenz von {F}olgen.
\newblock {\em Math. Scand.}, 23:169--170 (1969), 1968.

\bibitem{Glick}
Irving Glicksberg.
\newblock Uniform boundedness for groups.
\newblock {\em Canad. J. Math.}, 14:269--276, 1962.

\bibitem{HernMaca}
Salvador Hern{\'a}ndez and Sergio Macario.
\newblock Dual properties in totally bounded abelian groups.
\newblock {\em Arch. Math. (Basel)}, 80(3):271--283, 2003.

\bibitem{NeuWig}
J.~Neumann and E.~P. Wigner.
\newblock Minimally almost periodic groups.
\newblock {\em Ann. of Math. (2)}, 41:746--750, 1940.

\bibitem{Noble3}
N.~Noble.
\newblock The continuity of functions on {C}artesian products.
\newblock {\em Trans. Amer. Math. Soc.}, 149:187--198, 1970.

\bibitem{Noble}
N.~Noble.
\newblock $k$-groups and duality.
\newblock {\em Trans. Amer. Math. Soc.}, 151:551--561, 1970.

\bibitem{Reid}
G.~A. Reid.
\newblock On sequential convergence in groups.
\newblock {\em Math. Z.}, 102:227--235, 1967.

\bibitem{RossStro}
Kenneth~A. Ross and Karl Stromberg.
\newblock Baire sets and {B}aire measures.
\newblock {\em Ark. Mat.}, 6:151--160 (1965), 1965.

\bibitem{Varop}
N.~Th. Varopoulos.
\newblock A theorem on the continuity of homomorphisms of locally compact
  groups.
\newblock {\em Proc. Cambridge Philos. Soc.}, 60:449--463, 1964.

\end{thebibliography}
