Some applications of discrete selectivity and Banakh property in function spaces

作     者：Tkachuk, Vladimir V. 

作者机构：Univ Autonoma Metropolitana Dept Matemat Ave San Rafael Atlixco 186 Mexico City 09340 DF Mexico 

出 版 物：《EUROPEAN JOURNAL OF MATHEMATICS》 (Eur. J. Math.)

年 卷 期：2020年第6卷第1期

页      面：88-97页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Domination by a space Strong domination by a space Function spaces Lindelof  math> sigmami mathvariant="normal">sigma mi> math> documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{ oddsidemargin}{-69pt} begin{document}$$ Sigma $$ end{document}-space Metrizable space Banakh property Discrete selectivity Essentially uncountable space 

摘      要：We establish that an uncountable space X must be essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality ext(X)=t(X)=omega$$\mathrm{ext}(X)= t(X)=\omega $$\end{document} implies that the space Cp(X,[0,1]) is discretely selective. If X is a metrizable space, then Cp(X,[0,1])has the Banakh property if and only if so does Cp(Y,[0,1]) for some closed separable Y subset of XWe apply the above results to show that, for a metrizable X, the space Cp(X,[0,1])is strongly dominated by a second countable space if and only if X is homeomorphic to D circle plus M where D is a discrete space and M is countable. For a metrizable space X, we also prove that Cp(X,[0,1])has the Lindelof sigma-property if and only if the set of non-isolated points of X is second countable. Our results solve several open questions.