publication . Article . Preprint . 2018

Realizing spaces as path-component spaces

Banakh, Taras; Brazas, Jeremy;
Open Access
  • Published: 22 Mar 2018 Journal: Fundamenta Mathematicae, volume 248, pages 79-89 (issn: 0016-2736, eissn: 1730-6329, Copyright policy)
  • Publisher: Institute of Mathematics, Polish Academy of Sciences
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 grou...
free text keywords: Algebra and Number Theory, Topology, Mathematics, Algebra, Mathematics - General Topology, Mathematics - Algebraic Topology, 55Q52, 58B05, 54B15, 22A05, 54C10, 54G15
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