On non-cut points and COTS (Q1744607)

A connected ordered topological space (COTS) is a connected space \(X\) in which given any subset \(Y\) consisting of exactly three points, there is \(y\in Y\) such that \(Y\) meets both components of \(X\setminus \{y\}\) (no separation axiom is assumed). A subset \(Y\) of the space \(X\) is said to be \textsl{cut point convex} if whenever \(a,b\in Y\), then the subset \(S(a,b)\) of all points of \(X\) which separate \(X\) between \(a\) and \(b\) lies in \(Y\) also. A subset \(Y\) of \(X\) is said to be an \(H\)-\(set\) if every cover of \(Y\) by open subsets of \(X\) contains a finite family whose union is dense in \(Y\). This paper studies relations between cut point convex sets, \(H\)-sets and COTS. For example, among many other results it is shown that (1) If \(X\) is a COTS and \(H\) is an \(H\)-subset of \(X\), then \(H\subseteq S(a,b)\cup\{a,b\}\) for some \(a,b\in H\); and (2) If \(X\) is a connected space with at most two non-cut points such that every proper, non-degenerate regular closed connected subset \(K\) contains only a finite number of closed points of \(X\), then \(X\) is (homeomorphic to) a finite connected subspace of the Khalimsky line.