Realizing spaces as path-component spaces

TL;DR

This paper demonstrates that any Tychonoff space can be realized as the path-component space of another Tychonoff space with the same weight, and constructs a specific compact space with unusual fundamental group properties.

Contribution

It proves that every Tychonoff space is a path-component space of a Tychonoff space with matching weight and perfect quotient map, and constructs a compact space with a trivial shape homomorphism.

Findings

Every Tychonoff space is a path-component space of a Tychonoff space with the same weight.

Constructed a compact subset of rom which the fundamental group has specific complex properties.

The canonical homomorphism from the fundamental group to the shape group can be trivial in certain spaces.

Abstract

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.