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Compactness in function spaces with splitting topologies. (English) Zbl 1284.54030
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 \(C(X,Y)\).
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 \(C(X,Y)\).
MSC:
54C35 Function spaces in general topology
46Exx Linear function spaces and their duals
58D15 Manifolds of mappings
54C05 Continuous maps
54D30 Compactness
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