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RECOVERING THE BOUNDARY PATH SPACE OF A TOPOLOGICAL GRAPH USING POINTLESS TOPOLOGY

Part of: Basic constructions Topological dynamics Selfadjoint operator algebras Boolean algebras (Boolean rings) Distributive lattices

Published online by Cambridge University Press:  04 March 2020

GILLES G. DE CASTRO Open the ORCID record for GILLES G. DE CASTRO [Opens in a new window]
GILLES G. DE CASTRO*
Affiliation:
Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, 88040-970Florianópolis SC, Brazil e-mail: gilles.castro@ufsc.br
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Abstract

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

Keywords

topological graphsboundary path spacepointless topology

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 06D22: Frames, locales 37B10: Symbolic dynamics 54B05: Subspaces 54B10: Product spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 2 , October 2021 , pp. 232 - 248
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

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