Realizing spaces as path-component spaces

Abstract

The path component space of a topological space $X$ is the quotient space $π_0(X)$ 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 $Y\to π_0(Y)=X$ 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 $π_1(X,x_0)$ is an uncountable, cosmic, $k_ω$-topological group but for which the canonical homomorphism $ψ:π_1(X,x_0)\to \checkπ_1(X,x_0)$ to the first shape homotopy group is trivial.