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We prove that a 2-connected, locally connected, compact topological space M is homeomorphic to a subset of the 2-sphere if and only if M is metrizable and contains none of the Kuratowski graphs K5 and K3,3.
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...
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