Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Realizing spaces as path-component spaces   The path component space of a topological space $X$ is the quotient space
$\pi_0(X)$ whose points are the path components of $X$. We show that every
Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of
weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a
perfect map. Hence, many topological properties of $X$ transfer to $Y$. We
apply this result to construct a compact space $X\subset \mathbb{R}^3$ for
which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic,
$k_{\omega}$-topological group but for which the canonical homomorphism
$\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group
is trivial.

25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an