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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Tychonoff's theorem for locally compact spaces and an elementary approach to the topology of path spaces


Authors: Alan L. T. Paterson and Amy E. Welch
Journal: Proc. Amer. Math. Soc. 133 (2005), 2761-2770
MSC (2000): Primary 54B10, 46L05; Secondary 22A22, 46L85, 54B15
Published electronically: April 20, 2005
MathSciNet review: 2146225
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Abstract: The path spaces of a directed graph play an important role in the study of graph $C^*$-algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.


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Additional Information

Alan L. T. Paterson
Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
Email: mmap@olemiss.edu

Amy E. Welch
Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
Email: amy3welch@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08030-5
PII: S 0002-9939(05)08030-5
Keywords: Directed graphs, graph $C^*$-algebras, path spaces, Tychonoff's theorem
Received by editor(s): January 25, 2004
Published electronically: April 20, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society