If \(X\) is \(\sigma\)-compact Polish, then \(C_{k}\)(\(X\)) has a \(\sigma\)-closure-preserving base
type: Item
instance of: scholarly article
title: If \(X\) is \(\sigma\)-compact Polish, then \(C_{k}\)(\(X\)) has a \(\sigma\)-closure-preserving base
zbMATH Open document ID: 1074.54009
author: Gary Gruenhage
author: Ken-ichi Tamano
published in: Topology and its Applications
publication date: 2005-06-10T00:00:00Z
review text: As the title itself indicates, the main result of this paper is to show that if \(X\) is a \(\sigma\)-compact Polish space, then the function space \(C_k(X)\) of all continuous functions on \(X\) with the compact open topology is an M\(_1\)-space. First, the authors show that if \(X\) is such a space, then \(C_k(X)\) is a \(\mu\)-space. This gives a positive answer to a question by Gartside. Combining the result of Gartside and Reznichenko that \(C_k(X)\) is stratifiable if \(X\) is a Polish space with this new result, \(C_k(X)\) is automatically an M\(_1\)-space. However, the authors also construct directly a \(\sigma\)-closure-preserving base for \(C_k(X)\).
reviewed by: Takemi Mizokami
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 2174907
zbMATH Keywords: stratifiable
zbMATH Keywords: M\(_1\)-space
zbMATH Keywords: polish
zbMATH Keywords: function space
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.034
OpenAlex ID: W1980536431
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DOI: 10.1016/J.TOPOL.2003.08.034
description: scientific article; zbMATH DE number 2174907