We consider a number of very strong separation properties for connected spaces. A connected space X is called an Fsω-space provided that each separator between any two points of X contains a finite subset which is also a separator between these two points. If (X, T) is a connected, Hausdorff space in Fsω, then there exists a weaker topology F for X such that (X, F) is embedded in a continuum in Fsω. We give several characterizations for the class of Fsω-continua. A connected space is in Dsω if it is in Fsω and if every infinite subset without interior disconnects the space. We prove that if (X, T) is a separable, Hausdorff, Dsω-space then (X, F) is a metrizable, one-dimensional ANR with finitely generated fundamental group

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