Cut points in some connected topological spaces

摘要

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S|a,b| is compact whenever it is closed. Using this result we show that an H(ⅰ) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.