Every Čech-complete space is cofinally Baire

Abstract

We prove that any space X with a dense Čech-complete subspace is cofinally pseudocomplete, i.e., if $f:X\rightarrow M$ is a continuous onto map of X onto a second countable space M, then there exist continuous onto maps $g:X\rightarrow P$ and $h:P\rightarrow M$ such that $f=h\circ g$ while P is second countable and has a dense Polish subspace. We show that $C_p(X)$ is cofinally pseudocomplete if and only if it is pseudocomplete and $C_p(X,[0,1])$ 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 $C_p(X)$ 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.