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We show that a compactum is locally connected if and only if every semimonotone mapping onto it is also monotone. If we put open in place of monotone, we obtain the finite compacta; if we put in confluent, we obtain a large class of compacta, the connected members of which are connected im kleinen at each of their cut points.
Bankston, Paul, "Defining Topological Properties via Interactive Mapping Classes" (2009). Mathematics, Statistics and Computer Science Faculty Research and Publications. 2.