Realizing spaces as path-component spaces
Volume 248 / 2020
Fundamenta Mathematicae 248 (2020), 79-89
MSC: Primary 55Q52, 58B05, 54B15; Secondary 22A05, 54C10, 54G15.
DOI: 10.4064/fm529-12-2018
Published online: 28 June 2019
Abstract
The path-component space $\pi _0(X)$ of a topological space $X$ is the quotient space of $X$ 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 $T_4$ topological group but the canonical homomorphism $\psi :\pi _1(X,x_0)\to \check{\pi }_1(X,x_0)$ to the first shape homotopy group is trivial.
