Concerning Cut Point Spaces of Order Three

Abstract

A pointpof a topological spaceXis a cut point ofXifX−{p}is disconnected. Further, ifX−{p}has preciselymcomponents for some natural numberm≥2we will say thatphas cut point orderm. If each pointyof a connected spaceYis a cut point ofY, we will say thatYis a cut point space. Herein we construct a spaceSso thatSis a connected Hausdorff space and each point ofSis 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.