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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
##### Keywords:
function spaces; continuous functions; compactness
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