Polish factorizations, cosmic spaces and domain representability

Abstract

We say that a space $X$ is {\it cofinally Polish} if for every continuous onto map $f:X\to M$ of $X$ onto a separable metrizable space $M$, there exists a Polish space $P$ and continuous onto maps $g:X\to P$ and $h:P\to M$ such that $f=h\circ g$. We study general properties of cofinally Polish spaces and compare the property of being cofinally Polish with subcompactness and domain representability. It is established, among other things, that a space with a countable network is cofinally Polish if and only if it is domain representable. We also show that any $G_\delta$-subset of an Eberlein compact space must be subcompact thus giving an answer to an open problem published in 2013.