Polish factorizations, cosmic spaces and domain representability

We say that a space X is cofinally Polish if for every continuous onto map f : X -> M of X onto a separable metrizable space M, there exists a Polish space P and continuous onto maps g : X -> P and h : P -> M such that f = h o 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.