A Note on Connected Ordered Topological Spaces

Abstract

Some properties of Connected Ordered Topological Spaces (COTS) are obtained. We prove that if a connected space X has at most two non-cut points and an R(i) subset, the closure of which contains all closed points of X, then X is a COTS with endpoints. The concept of locally cut point convex topological space is introduced. It is proved that in a connected and locally cut point convex space, for $a, b$ in $X$, $S[a, b]$ is compact whenever it is closed.