Tychonoff’s theorem for locally compact spaces and an elementary approach to the topology of path spaces
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- by Alan L. T. Paterson and Amy E. Welch PDF
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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.References
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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
- Received by editor(s): January 25, 2004
- Published electronically: April 20, 2005
- Communicated by: David R. Larson
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2761-2770
- MSC (2000): Primary 54B10, 46L05; Secondary 22A22, 46L85, 54B15
- DOI: https://doi.org/10.1090/S0002-9939-05-08030-5
- MathSciNet review: 2146225