Domain representability of certain function spaces
type: Item
instance of: scholarly article
title: Domain representability of certain function spaces
zbMATH Open document ID: 1170.54006
published in: Topology and its Applications
publication date: 2009-07-01T00:00:00Z
full work available at URL: https://scholarworks.wm.edu/cgi/viewcontent.cgi
review text: A topological space is domain representable if it is homeomorphic to the subspace of maximal elements of a domain, topologized with the Scott topology. Continuing their systematic investigation of domain representability, the authors study this property in the space \(C_p(X)\) of continuous real-valued functions with the topology of pointwise convergence. Motivated by a result of Tkachuk on subcompactness of \(C_p(X)\), they conjecture that for a Tychonoff space \(X\), \(C_p(X)\) is domain representable iff \(X\) is discrete. The main theorem of the paper is a partial result confirming the conjecture for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. As domain representability is closely related to various completeness properties associated with topological games, the authors also characterize the effect of these games on \(C_p(X)\) for a completely regular pseudonormal \(X\). One of the completeness properties studied is strong Choquet completeness which is associated with the so-called strong Choquet game. As a corollary of previous results, it is shown that there exists a GO-space \(X\) such that \(C_p(X)\) is strongly Choquet complete and yet it is not domain representable.
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 5573805
zbMATH Keywords: domain representable space
zbMATH Keywords: function space
zbMATH Keywords: pointwise convergence topology
zbMATH Keywords: pseudo-radial space
zbMATH Keywords: transfinite sequence
zbMATH Keywords: pseudocomplete
zbMATH Keywords: weakly Choquet complete
zbMATH Keywords: strongly Choquet complete
author: Harold Bennett
author: David Lutzer
reviewed by: Laszlo Zsilinszky
MaRDI profile type: Publication
OpenAlex ID: W2050448654
cites work: Domain-representable spaces
cites work: Q5571891
cites work: Q5728951
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cites work: Topological games in domain theory
cites work: Q4794155
cites work: Topological properties of spaces of continuous functions
cites work: Cartesian products of Baire spaces
cites work: Baire property of spaces of continuous functions
DOI: 10.1016/J.TOPOL.2009.03.013
description: scientific article; zbMATH DE number 5573805