Classification Of Locally 2-Connected Compact Metric Spaces

The aim of this paper is to prove that, for compact metric spaces which do not contain infinite complete graphs, the (strong) property of being “locally 2-dimensional” is guaranteed just by a (weak) local connectivity condition. Specifically, we prove that a locally 2-connected, compact metric space M either contains an infinite complete graph or is surface like in the following sense: There exists a unique surface S such that S and M contain the same finite graphs. Moreover, M is embeddable in S, that is, M is homeomorphic to a subset of S.

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Correspondence to Carsten Thomassen.

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Thomassen, C. Classification Of Locally 2-Connected Compact Metric Spaces. Combinatorica 25, 85–103 (2004).

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Mathematics Subject Classification (2000):

  • 05C10
  • 57M15