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Closure-preserving covers in function spaces

David Guerrero Sánchez — 2010

Commentationes Mathematicae Universitatis Carolinae

It is shown that if C p ( X ) admits a closure-preserving cover by closed σ -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.

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