The Ascoli property for function spaces

详细信息    查看全文

• 作者：Saak Gabriyelyan^a ; ^{saak@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Jan Grebí ; k^b ; ¹ ; ^{Greboshrabos@seznam.cz" class="auth_mail" title="E-mail the corresponding author} ; Jerzy Ka̧kol^c ; ^b ; ² ; ^{kakol@amu.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Lyubomyr Zdomskyy^d ; ³ ; ^{lzdomsky@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54C35 ; 54D50
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：214
• 期：Complete
• 页码：35-50
• 全文大小：440 K

文摘

The paper deals with Ascoli spaces cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ and C_k(X)$C_k(X)$ over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X   for which any compact subset K$\mathcal{K}$ of C_k(X)$C_k(X)$ is evenly continuous, essentially includes the class of k_R$k_{\mathbb{R}}$-spaces. First we prove that if cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Ascoli, then it is κ-Fréchet–Urysohn. If X   is cosmic, then cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Ascoli iff it is κ-Fréchet–Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space X   the space cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Ascoli, then X is scattered. If X   is scattered and stratifiable, then cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is an Ascoli space. Consequently: (a) If X   is a complete metrizable space, then cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Ascoli iff X is scattered. (b) If X   is a Čech-complete Lindelöf space, then cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Ascoli iff X   is scattered iff cd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)$C_p(X)$ is Fréchet–Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent: (i) X   is locally compact. (ii) C_k(X)$C_k(X)$ is a k_R$k_{\mathbb{R}}$-space. (iii) C_k(X)$C_k(X)$ is an Ascoli space. The Ascoli spaces C_k(X,I)$C_k(X,\mathbb{I})$ are also studied.