Every Čech-complete space is cofinally Baire (Q6552639)

Given a topological property \(\mathcal{P}\), a space \(X\) is cofinally \(\mathcal{P}\) if for any continuous onto map \(f:X\to M\) of \(X\) onto a second countable space \(M\), there exist continuous onto maps \(g:X\to P\) and \(h:P\to M\) such that \(f=h\circ g\), the space \(P\) is second countable and has the property \(\mathcal{P}\). The authors study cofinal properties in some nice classes such as pseudocompleteness, local compacteness, the Baire property, etc.\N\NA space \(X\) is called \textit{pseudocomplete} if there exists a sequence \(\{\mathcal{B}_n:n\in\omega\}\) of \(\pi\)-bases in \(X\) such that a family \(\{U_n:n\in\omega\}\) of non-empty open subsets of \(X\) has non-empty intersection whenever \(U_n\in\mathcal{B}_n\) and \(\overline{U}_{n+1}\subset U_n\) for every \(n\in\omega\). A metrizable space is pseudocomplete if and only if has a dense Čech-complete subspace. The authors also characterize cofinal pseudocompleteness in spaces \(C_p(X)\) and \(C_p(X,[0,1])\). Then they 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 if it is pseudocompact. Furthemore, they prove the space \(C_p(X)\) is cofinally locally compact if and only is \(X\) is finite.\N\NIt is still an open question whether every Baire space must be cofinally Baire. However the authors establish as a main result in this paper that every Čech-complete space is cofinally pseudocomplete and hence cofinally Baire. Besides, any locally countably compact GO space of countable extent is cofinally locally compact and hence cofinally Polish.\N\NThe results of this article resolve several published open questions, and at the end of the paper a list of other open questions has also been proposed.