End spaces and spanning trees

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• 作者：Reinhard Diestel
• 关键词：Trees ; Ends of graphs ; Metrizability ; Freudenthal compactification ; Normal spanning trees
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2006
• 出版时间：November 2006
• 年：2006
• 卷：96
• 期：6
• 页码：846-854
• 全文大小：124 K

文摘

We determine when the topological spaces |G| naturally associated with a graph G and its ends are metrizable or compact.

In the most natural topology, |G| is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone's theorem that metric spaces are paracompact.

We show that e61cb5b000808a"" title=""Click to view the MathML source"">|G| is compact in the most natural topology if and only if no finite vertex separator of G leaves infinitely many components. When G is countable and connected, this is equivalent to the existence of a locally finite spanning tree. The proof uses ultrafilters and a lemma relating ends to directions.