Weakly connected subsets (Q1026907)

A subset \(S\) of a topological space \((X,\tau)\) is said to be \textsl{weakly connected in} \(X\) if whenever \(U\) is an open and closed subset of \(X\), then either \(S\subseteq U\) or \(S\subseteq X\setminus U\). As such, being a weakly connected subset is not a topological property but depends on the embedding. The author obtains some characterizations and simple properties of weakly connected subsets. A space whose quasicomponents differ from its components gives a negative answer to Problem 1.