# Article

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Keywords:
closure-preserving covers; function spaces; compact spaces; pointwise convergence topology; topological game; winning strategy
Summary:
It is shown that if $C_p(X)$ admits a closure-preserving cover by closed $\sigma$-compact sets then $X$ is finite. If $X$ is compact and $C_p(X)$ has a closure-preserving cover by separable subspaces then $X$ is metrizable. We also prove that if $C_p(X,[0,1])$ has a closure-preserving cover by compact sets, then $X$ is discrete.
References:
[1] Arkhangelskii A.V.: Topological function spaces. Mathematics and its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1144519
[2] Junnila H.J.K.: Metacompactness, paracompactness, and interior-preserving open covers. Trans. Amer. Math. Soc. 249 (1979), no. 2, 373–385. DOI 10.1090/S0002-9947-1979-0525679-9 | MR 0525679 | Zbl 0404.54017
[3] Engelking R.: General Topology. Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[4] Potoczny H.B.: Closure-preserving families of compact sets. General Topology and Appl. 3 (1973), 243–248. DOI 10.1016/0016-660X(72)90015-3 | MR 0322805 | Zbl 0263.54010
[5] Potoczny H.B., Junnila H.J.K.: Closure-preserving families and metacompactness. Proc. Amer. Math. Soc. 53 (1975), no. 2, 523–529. DOI 10.1090/S0002-9939-1975-0388337-1 | MR 0388337 | Zbl 0318.54018
[6] Rogers C.A., Jayne J.E., $K$-analytic Sets, Rogers C.A., Jayne J.E.: et al. “Analytic Sets", Academic Press, London, 1980, pp. 2–181.
[7] Shakmatov D.B., Tkachuk V.V.: When is the space $C_p(X)$ $\sigma$-countably compact?. Vestnik Moskov. Univ. Mat. 41 (1986), no. 1, 73–75.
[8] Telgársky R.: Spaces defined by topological games. Fund. Math. 88 (1975), no. 3, 193–223. MR 0380708
[9] Tkachuk V.V.: The spaces $C_p(X)$: decomposition into a countable union of bounded subspaces and completeness properties. Topology Appl. 22 (1986), no. 3, 241–253. DOI 10.1016/0166-8641(86)90023-4 | MR 0842658
[10] Tkachuk V.V.: The decomposition of $C_p(X)$ into a countable union of subspaces with “good” properties implies “good” properties of $C_p(X)$. Trans. Moscow Math. Soc. 55 (1994), 239–248. MR 1468461

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