A Note on Non-cut Points, VH-sets, Almost VH-sets and COTS

Abstract

Two concepts of VH-sets and almost VH-sets are introduced to study connected ordered topological spaces (COTS). We show that every $R(i)$ subset of a connected space is an almost VH-set. Two characterizations of COTS with endpoints using the concept of almost VH-set have been obtained. It is also proved that if $X$ is a connected non-cut point inclined space and the removal of any two-point disconnected set of it leaves the space disconnected, then each one of $H \cup \{a, b\}$ and $K \cup \{a, b\}$ has exactly two non-cut points, and is homeomorphic to a finite connected subspace of the Khalimsky line where $H$ and $K$ are separating sets of $X \setminus \{a, b\}$, $a, b \in X$.