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.