The Baire property in the compact-open topology of Lašnev spaces (Q260578)

If \(X\) is a topological space and we have collections \(\mathcal A\) and \(\mathcal B\) of subsets of \(X\), then \textit{\(\mathcal A\) moves off \(\mathcal B\)} if for any \(B\in \mathcal B\), there is \(A\in \mathcal A\) such that \(A\cap B=\emptyset\). A collection \(\mathcal C\) of subsets of \(X\) is \textit{strongly discrete} if there exists a discrete family \(\{U_C: C\in \mathcal C\}\) of open subsets of \(X\) such that \(C\subset U_C\) for every \(C\in \mathcal C\). Let \(\mathcal K(X)\) be the family of all compact subsets of \(X\).NEWLINENEWLINEIt is said that a space \(X\) has \textit{the moving off property} if any family of compact subsets of \(X\) that moves off \(\mathcal K(X)\) has an infinite strongly discrete subfamily. The expression \(C_k(X)\) stands for the set of all real-valued continuous functions on \(X\) endowed with the compact-open topology.NEWLINENEWLINEThe author proves, among other things, that for any first countable paracompact space \(X\), if \(Y\) is a closed continuous image of \(X\) with the moving off property, then \(Y\) is locally compact. He deduces from this fact that for any closed image \(Y\) of a first countable paracompact space, the Baire property of the space \(C_k(Y)\) is equivalent to the moving off property of \(Y\).