An extension theorem for strongly locally homogeneous continua (Q1957136)

The main result: If \(X\) is a separable metrizable continuum that is strongly locally homogeneous and is not homeomorphic to \(S^1\) then every homeomorphism between two countable compact subsets of \(X\) can be extended to a homeomorphism on \(X\). (A space \(X\) is said to be strongly locally homogeneous if every neighborhood \(U\) of \(x\) contains an open neighborhood \(V\) of \(x\) such that for every \(y\in V\) there is a homeomorphism \(f\) on \(X\) taking \(y\) to \(x\) and leaving points of \(X\setminus U\) fixed.) The result extends similar results for manifolds.