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

Finding an internal condition on a topological space X   which is necessary and sufficient for the compact-open topology Ck(X)Ck(X) to be Baire is an open problem. The moving off property is a known characterization for the class of first-countable or locally compact spaces. Here we show that it also holds for the class of closed images of first-countable paracompact spaces, hence Lašnev spaces in particular; furthermore, Baireness of Ck(X)Ck(X) is equivalent to its α-favorability for X in this class. We will also show that if X   is the closed image of a locally compact paracompact space then Ck(X)Ck(X) is α-favorable.