The Ascoli property for function spaces

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• 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si2.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d6257d33c3e11570108c282b06601b00" title="Click to view the MathML source">C_k(X)class="mathContainer hidden">class="mathCode">$C_k(X)$ over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X   for which any compact subset class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si3.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=1ecbcc1246db2522c77d3d91473120ac" title="Click to view the MathML source">Kclass="mathContainer hidden">class="mathCode">$\mathcal{K}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si2.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d6257d33c3e11570108c282b06601b00" title="Click to view the MathML source">C_k(X)class="mathContainer hidden">class="mathCode">$C_k(X)$ is evenly continuous, essentially includes the class of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si13.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d08d0bb7863ae89eb45d2378981b4bcf" title="Click to view the MathML source">k_Rclass="mathContainer hidden">class="mathCode">$k_{\mathbb{R}}$-spaces. First we prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ is Ascoli, then it is κ-Fréchet–Urysohn. If X   is cosmic, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ is Ascoli, then X is scattered. If X   is scattered and stratifiable, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ is an Ascoli space. Consequently: (a) If X   is a complete metrizable space, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ is Ascoli iff X is scattered. (b) If X   is a Čech-complete Lindelöf space, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ is Ascoli iff X   is scattered iff class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si1.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=3fefcd461d3028ab5aa08b376d558871" title="Click to view the MathML source">C_p(X)class="mathContainer hidden">class="mathCode">$C_p(X)$ 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) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si2.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d6257d33c3e11570108c282b06601b00" title="Click to view the MathML source">C_k(X)class="mathContainer hidden">class="mathCode">$C_k(X)$ is a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si13.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d08d0bb7863ae89eb45d2378981b4bcf" title="Click to view the MathML source">k_Rclass="mathContainer hidden">class="mathCode">$k_{\mathbb{R}}$-space. (iii) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si2.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=d6257d33c3e11570108c282b06601b00" title="Click to view the MathML source">C_k(X)class="mathContainer hidden">class="mathCode">$C_k(X)$ is an Ascoli space. The Ascoli spaces class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116302152&_mathId=si19.gif&_user=111111111&_pii=S0166864116302152&_rdoc=1&_issn=01668641&md5=ca3d128c2ddc5db5699627e14ba07310" title="Click to view the MathML source">C_k(X,I)class="mathContainer hidden">class="mathCode">$C_k(X,\mathbb{I})$ are also studied.