International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 10679, 8 pages
doi:10.1155/2007/10679
Research Article
Concerning Cut Point Spaces of Order Three
1Department of Mathematics, Lamar University, Beaumont 77710, TX, USA
2Department of Mathematics and Computer Science, Emory University, Atlanta 30322, GA, USA
Received 15 April 2005; Accepted 15 May 2007
Academic Editor: Gerald F. Jungck
Copyright © 2007 D. Daniel and William S. Mahavier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.