Concerning Cut Point Spaces of Order Three

Abstract

A point p of a topological space X is a cut point of X if X−{p} is disconnected. Further, if X−{p} has precisely m components for some natural number m≥2 we will say that p has cut point order m. If each point y of a connected space Y is a cut point of Y, we will say that Y is a cut point space. Herein we construct a space S so that S is a connected Hausdorff space and each point of S is a cut point of order three. We also note that there is no uncountable separable cut point space with each point a cut point of order three and therefore no such space may be embedded in a Euclidean space.