On the existence of free topological groups
type: Item
instance of: scholarly article
title: On the existence of free topological groups
zbMATH Open document ID: 0649.22001
DOI: 10.1016/0166-8641(88)90024-7
author: Jan van Mill
author: W. W. Comfort
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: For a class U of topological groups (such as for instance the class consisting of the circle group), and for a space X which is homeomorphic to a subspace of a product of factors selected from U, the authors construct a topological group G containing a copy of X such that any continuous function \(f: X\to H\) for a U-group H (that is, a group isomorphic to a closed subgroup of a product of U-groups) has a unique extension to a continuous group homomorphism \(G_ U\to H\). If G is itself isomorphic to a closed subgroup of a product of U-groups, then they call G the free U-group \(G_ U\). (They offer a variation involving a second class V of topological groups for the picking of H. This gives the (U,V)-free groups.) The authors show that for the class U of pseudocompact groups, a free U- group exists if and only if X is empty. They then turn to the case \(U=\{{\mathbb{R}}/{\mathbb{Z}}\}\), that is, to the case of free compact abelian groups. In this case one obtains FX, the character group Hom(C(X,\({\mathbb{R}}/{\mathbb{Z}}),{\mathbb{R}}/{\mathbb{Z}})\). The authors call PX the subgroup of all characters with countable range and define the group \(\xi\) X as the subgroup generated in \(F\beta\) X by the set \(X\cup P\beta X\). This turns out to be a (PA,CA)-free group for the class of pseudocompact, respectively, compact abelian groups such that X is embedded as a closed subspace. The authors construct other groups \(\eta\) X for certain spaces with properties very similar to those of \(\xi\) X but which allows no nonconstant continuous homomorphism \(\xi\) \(X\to \eta X\). The paper concludes with a list of unsolved problems.
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 20K45
zbMATH DE Number: 4059722
zbMATH Keywords: topological groups
zbMATH Keywords: free groups
zbMATH Keywords: pseudocompact groups
zbMATH Keywords: free compact abelian groups
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90024-7
OpenAlex ID: W2121131932
cites work: Q5586466
cites work: Q3795895
cites work: Pseudocompactness and uniform continuity in topological groups
cites work: Countably compact groups and finest totally bounded topologies
cites work: Normality in Subsets of Product Spaces
cites work: The Product of Two Countably Compact Topological Groups
cites work: Q3255890
cites work: On epi-reflective hulls
cites work: Q5731806
cites work: Stone-Cech Compactifications of Products
cites work: Q5802187
cites work: E-kompakte Räume. (E-compact spaces)
cites work: Q5534000
cites work: Q5723435
cites work: Free compact groups. I: Free compact abelian groups
cites work: Correction to ``Free compact groups. I: Free compact abelian groups''
cites work: Free topological groups and infinite direct product topological groups
cites work: Adjoint Functors
cites work: Reflective Functors in General Topology and Elsewhere
cites work: Uniform Continuity and Compactness in Topological Groups
cites work: FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES
cites work: Q5508993
cites work: Varieties of topological groups and left adjoint functors
cites work: Varieties of topological groups a survey
cites work: Q3699945
cites work: Note on free topological groups
cites work: On universal mappings and free topological groups
cites work: Q5528440
cites work: Q5522742
cites work: Free topological groups
description: scientific article; zbMATH DE number 4059722