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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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[1] Di Maio, G.; Holá, L’.; Holý, D.; McCoy, R.A., Topologies on the space of continuous functions, Topology appl., 86, 105-122, (1998) · Zbl 0940.54023
[2] Engelking, R., General topology, (1989), PWN Warsaw, rev. completed ed · Zbl 0684.54001
[3] Gillman, L.; Jerison, M., Rings of continuous functions, (1976), Springer New York · Zbl 0151.30003
[4] Hewitt, E., Rings of real-valued continuous functions I, Trans. amer. math. soc., 48, 64, 54-99, (1948) · Zbl 0032.28603
[5] Hirsch, M., Differential topology, (1976), Springer Berlin · Zbl 0356.57001
[6] Krikorian, N., Compositio math., 21, 343-348, (1969)
[7] I. Kupka, Convergences preserving the fixed point property, Math. Slovaca, to appear
[8] McCoy, R.A., Fine topology on function spaces, Internat. J. math. sci., 9, 417-424, (1978) · Zbl 0614.54014
[9] McCoy, R.A.; Ntantu, I., Topological properties of spaces of continuous functions, Lecture notes in math., 1315, (1988), Springer Berlin
[10] Naimpally, S.A., Graph topology for function spaces, Trans. amer. math. soc., 123, 267-271, (1966) · Zbl 0151.29703
[11] Spring, D., Compactness in the fine topology, Topology appl., 18, 87-94, (1984) · Zbl 0537.54006
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