Geodesic
Closure-preserving covers in function spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 693-703.

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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.
Classification : 54C35
Keywords: closure-preserving covers; function spaces; compact spaces; pointwise convergence topology; topological game; winning strategy 
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     title = {Closure-preserving covers in function spaces},
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Sánchez, David Guerrero. Closure-preserving covers in function spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 693-703. http://geodesic.mathdoc.fr/item/CMUC_2010__51_4_a11/