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Function spaces of completely metrizable spaces. (English) Zbl 0841.54012
Summary: Let $$X$$ and $$Y$$ be metric spaces and let $$\varphi : C_p (X) \to C_p (Y)$$ (resp. $$\varphi : C_p^* (X) \to C_p^* (Y))$$ be a continuous linear surjection. We prove that $$Y$$ is completely metrizable whenever $$X$$ is. As a corollary we obtain that complete metrizability is preserved by $$l_p$$- (resp. $$l^*_p$$-)equivalence in the class of all metric spaces. This solves Problem 35 in [Problems in $$C_p$$-theory, in ‘Open problems in topology’ (1990; Zbl 0718.54001)] (raised by Arkhangel’skij).

##### MSC:
 54C35 Function spaces in general topology 57N17 Topology of topological vector spaces
##### Keywords:
completely metrizable spaces; complete metrizability
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##### References:
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