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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)

##### MSC:
 54C35 Function spaces in general topology 54D30 Compactness 54C05 Continuous maps 54E35 Metric spaces, metrizability
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##### References:
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