Every Čech-complete space is cofinally Baire

期刊论文
发表日期：

2024年5月9日

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V. V. Tkachuk,
R. G. Wilson

摘要

We prove that any space X with a dense Čech-complete subspace is cofinally pseudocomplete, i.e., if $f:X\rightarrow M$ $f : X \to M$ is a continuous onto map of X onto a second countable space M, then there exist continuous onto maps $g:X\rightarrow P$ $g : X \to P$ and $h:P\rightarrow M$ $h : P \to M$ such that $f=h\circ g$ $f = h \circ g$ while P is second countable and has a dense Polish subspace. We show that $C_p(X)$ $C_{p} (X)$ is cofinally pseudocomplete if and only if it is pseudocomplete and $C_p(X,[0,1])$ $C_{p} (X, [0, 1])$ is cofinally pseudocomplete if and only it is pseudocompact. We introduce, in an analogous way, the class of cofinally locally compact spaces and show that $C_p(X)$ $C_{p} (X)$ is cofinally locally compact if and only if X is finite. Besides, any locally countably compact GO space of countable extent is cofinally locally compact and hence cofinally Polish. Our results solve several published open questions.