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Compactness in the fine and related topologies. (English) Zbl 0990.54016
Let \(X\) be a Tikhonov space, \(Y\) a metrizable space and \(C(X,Y)\) be the space of continuous functions from \(X\) to \(Y\). In [ibid. 18, 89-94 (1984; Zbl 0537.54006)] D. Spring studied the compactness of a space \(C(X,Y)\) with the fine topology. In this paper, the authors are interested in compactness of subsets of \(C(X,Y)\) with respect to the fine, Krikorian and graph topologies. For a paracompact, locally hemicompact \(k\)-space \(X\), let \(\tau\) be one of the following topologies: fine, graph, or Krikorian, the authors show that the following are equivalent for a subset \(Q\) of \(C(X,Y)\): (1) \(Q\) is countably compact in \((C(X,Y)\), \(\tau)\); (2) \(Q\) is compact in \((C(X, Y),\tau)\); (3) \(Q\) is sequentially compact in \((C(X,Y)),\tau)\); (4) \(Q\) is almost compactly supported and \(Q\) is compact in \((C(X,Y),\tau_{\text{co}})\), where \(\tau_{\text{co}}\) is the comapct-open topology. The results greatly generalize Spring’s results.
Reviewer: Shou Lin (Fujian)

54C35 Function spaces in general topology
54D30 Compactness
54C05 Continuous maps
54E35 Metric spaces, metrizability
Full Text: DOI
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