Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Families of retractions and families of closed subsets on compact spaces   It is know that the Valdivia compact spaces can be characterized by a special
family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know
that there are compact spaces with $r$-skeletons which are not Valdivia. In
this paper, we shall study $r$-squeletons and special families of closed
subsets of compact spaces. We prove that if $X$ is a zero-dimensional compact
space and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ such that $|r_s(X)|
\leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of
isolated points. Also we give conditions to an $r$-skeleton in order that this
$r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of
the base space. The standard definition of a Valdivia compact spaces is via a
$\Sigma$-product of a power of the unit interval. Following this fact we
introduce the notion of $\pi$-skeleton on a compact space $X$ by embedding $X$
in a suitable power of the unit interval together with a pair
$(\mathcal{F},\varphi)$, where $\mathcal{F}$ is family of metric separable
subspaces of $X$ and $\varphi$ an $\omega$-monotone function which satisfy
certain properties. This new notion generalize the idea of a $\Sigma$-product.
  We prove that a compact space admits a retractional-skeleton iff it admits a
$\pi$-skeleton. This equivalence allows to give a new proof of the fact that
the product of compact spaces with retractional-skeletons admits an
retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact
spaces are characterized by a special family of closed subsets. Following this
direction, we introduce the notion of weak $c$-skeleton which under certain
conditions characterizes the Valdivia compact spaces and compact spaces with
$r$-skeletons.

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The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom