Baireness of \(C_ k(X)\) for locally compact \(X\) (Q1371933)

An unsolved problem is to characterize in terms of \(X\) when the space \(C_k(X)\) of continuous real-valued functions on \(X\) with the compact-open topology is a Baire space. This paper extends the known partial results on this problem to give a solution in the case that \(X\) is a q-space (which includes locally compact \(X\) and first countable \(X\)). In this case, the property on \(X\) that is equivalent to \(C_k(X)\) being a Baire space is called the moving off property (MOP); which means that for every collection \(\mathcal K\) of nonempty compact subsets of \(X\) such that every compact subset of \(X\) is disjoint from some member of \(\mathcal K\), there is an infinite subcollection of \(\mathcal K\) having a discrete open expansion in \(X\). A characterization of MOP is also given in terms of a topological game. Other results include: every pseudocompact space with MOP is compact (hence \(C_k(\omega_1)\) is not a Baire space), and every q-space with MOP is locally compact. Finally, an example is given in ZFC of a locally compact space \(X\) such that \(C_k(X)\) is a Baire space but is not weakly \(\alpha\)-favorable.