Título: Compact covers and function spaces

Autor: Kakol, J.; López Pellicer, Manuel; Okunev, O.

Resumen: For a Tychonoff space X, we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelof $Sigma$-property of X in terms of Cp(X), we extend Okunev's results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then $nu$Y is a Lindelof $Sigma$-space (respectively K-analytic) if $nu$X has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space.We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by lp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and lp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.

URI: http://hdl.handle.net/10251/56861

Fecha: 2014-03-01

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