Compactness in function spaces with splitting topologies (Q2871482)

Let \(C(X,Y)\) be the set of all continuous functions from a topological space \(X\) into a topological space \(Y\) and let \(H\) be a subset of \(C(X,Y)\).NEWLINENEWLINE In the paper under review the authors consider certain topologies \(\tau_{\mathfrak A}\) of \(\mathfrak A\)-convergence on \(C(X,Y)\) which are finer than the topology \(\tau_p\) of pointwise convergence. Their main result is a criterion for the \(\tau_{\mathfrak A}\)-compactness of \(H\) in \(C(X,Y)\), whose validity requires the assumption that \(H\) is evenly continuous on each \(B\in \mathfrak A\). It follows from what is discussed in the paper that, if \(Y\) is a Hausdorff space, then \(H\) is \(\tau_{co}\)-compact in \(C(X,Y)\) (\(\tau_{co}\) being the compact-open topology) if and only if \(H(x)\) is relatively compact in \(Y\) for all \(x\in X\), \(H\) is evenly continuous on each compact subset of \(X\), and \(H\) is \(\tau_p\)-closed in \(C(X,Y)\).