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

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...

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Published inJournal of the Australian Mathematical Society (2001) Vol. 111; no. 2; pp. 232 - 248
Main Author DE CASTRO, GILLES G.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2021
Subjects
Topology
54B10
boundary path space
06D22
37B10
pointless topology
topological graphs
54B05
46L55
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ISSN1446-7887
1446-8107
DOI10.1017/S1446788720000051

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Summary: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.
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ISSN:1446-7887
1446-8107
DOI:10.1017/S1446788720000051