A study of cut points in connected topological spaces

We prove that every H ( i ) subset H of a connected space X such that there is no proper connected subset of X containing H, contains at least two non-cut points of X. This is used to prove that a connected space X is a COTS with endpoints if and only if X has at most two non-cut points and has an H ( i ) subset H such that there is no proper connected subset of X containing H. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.