Generalization of the Comfort-Ross theorem. I
type: Item
instance of: scholarly article
title: Generalization of the Comfort-Ross theorem. I
zbMATH Open document ID: 0698.22004
DOI: 10.1007/BF01060321
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: A subspace X of a topological space Y is said to be bounded in Y if every continuous, real-valued function on Y is bounded on X. Using the elegant observation that a necessary and sufficient condition for boundedness of X in Y is that only finitely many elements of any locally finite family of sets open in Y can intersect X, it is shown that the cartesian product of an arbitrary family of bounded subsets of topological groups is bounded in the product of the groups. [This, of course, generalizes the well-known theorem of Comfort and Ross.] It is also shown that a continuous pseudometric on a topological group is uniformly continuous on any bounded subspace, as is any real-valued continuous function defined on G. Furthermore, the universal, two-sided, left and right uniformities from the group are equal on X. If G is algebraically generated by X and X is bounded, then any real-valued function on X can be factored through a continuous homomorphism onto a group of countable weight. A union of countably many bounded subspaces is said to be sigma-bounded. A (Tychonoff) space X is then sigma-bounded iff all of the following are also sigma-bounded: the free topological group on X, the free topological Abelian group on X, and the free linear locally convex space on X.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 4144285
zbMATH Keywords: continuous, real-valued function
zbMATH Keywords: locally finite family of sets
zbMATH Keywords: bounded subsets of topological groups
zbMATH Keywords: continuous pseudometric
zbMATH Keywords: uniformly continuous
zbMATH Keywords: left and right uniformities
zbMATH Keywords: continuous homomorphism
zbMATH Keywords: countable weight
zbMATH Keywords: sigma-bounded
zbMATH Keywords: free topological group
author: Mikhail Tkachenko
reviewed by: Douglas L. Grant
MaRDI profile type: Publication
cites work: Pseudocompactness and uniform continuity in topological groups
cites work: The Product of Two Countably Compact Topological Groups
cites work: REAL FUNCTIONS AND CANONICAL SETS IN TIKHONOV PRODUCTS AND TOPOLOGICAL GROUPS
cites work: Q3935830
cites work: Q3671393
description: scientific article; zbMATH DE number 4144285