Compact covers and function spaces |
| |
Authors: | J Ka?kol M Lopez-Pellicer O Okunev |
| |
Institution: | 1. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland;2. Depto. de Matemática Aplicada and IUMPA, Universitat Politècnica de València, E-46022 València, Spain;3. Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Rio Verde s/n, Col. San Manuel, Ciudad Universitaria, CP 72570 Puebla, Puebla, Mexico |
| |
Abstract: | 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 Lindelöf Σ-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 υY is a Lindelöf Σ-space (respectively K-analytic) if υX 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 ?p-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and ?p-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented. |
| |
Keywords: | Compact resolution C(X)- and Lp(X)- and Lp(X)-spaces" target="_blank">gif" overflow="scroll">Lp(X)-spaces ?ech-complete space Lindelö f Σ K-analytic analytic spaces ?p-" target="_blank">gif" overflow="scroll">?p- ?c- and t-equivalence" target="_blank">gif" overflow="scroll">?c- and t-equivalence Pointwise countable type spaces Polish space Real compactification μ-space Web-bounded spaces |
本文献已被 ScienceDirect 等数据库收录! |
|