End compactifications in non-locally-finite graphs

Details End compactifications in non-locally-finite graphs End compactifications in non-locally-finite graphs

2000-10-03

arxiv.org/abs/math/0010033v2

Mathematics

Combinatorics

17 pages, to appear in Math. Proc. Cambridge Philos. Soc

Scientific paper

10.1017/S0305004101005321

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in \cite{cartwright93martin}, which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.

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