Projective \(\sigma\)-compactness, \(\omega_1\)-caliber, and \(C_p\)-spaces
type: Item
instance of: scholarly article
title: Projective \(\sigma\)-compactness, \(\omega_1\)-caliber, and \(C_p\)-spaces
zbMATH Open document ID: 0944.54012
DOI: 10.1016/S0166-8641(99)00011-5
published in: Topology and its Applications
publication date: 2000-08-13T00:00:00Z
review text: Let \(C_p(X)\) be the space of continuous real-valued functions on \(X\) with the topology of pointwise convergence. A space is projectively \(\sigma\)-compact provided that every separable metrizable continuous image of the space is \(\sigma\)-compact. This article shows that \(C_p(X)\) is projectively \(\sigma\)-compact if and only if it is \(\sigma\)-pseudocompact. In addition, characterizations are given for \(C_p(X)\) to be projectively analytic and for \(C_p(Y|X)\) to be \(\sigma\)-compact, where \(C_p(Y|X)\) consists of the restrictions of the functions in \(C_p(X)\) to \(Y\). In the last section, a known result concerning spaces with cardinality no greater than \(2^{\omega}\) is generalized and used to show that the weight of a compact subspace of \(C_p(X)\) is no greater than \(2^{\omega}\) whenever \(\omega_1\) is a caliber of \(X\) and \(X\) has countable extent. This is a ZFC result so that it is related to the question of whether it is true in ZFC that every compact subspace of \(C_p(X)\) is metrizable whenever \(\omega_1\) is a caliber of \(X\).
reviewed by: R. A. McCoy
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 1462690
zbMATH Keywords: Hurewicz space
zbMATH Keywords: Lindelöf space
zbMATH Keywords: density
zbMATH Keywords: weight
zbMATH Keywords: analytic space
zbMATH Keywords: \(K\)-analytic space
author: A. V. Arhangel'skii
MaRDI profile type: Publication
full work available at URL: s0166-8641(99)00011-5
OpenAlex ID: W2027372250
description: scientific article; zbMATH DE number 1462690