Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 310832, 3 pages
doi:10.1155/2009/310832
Research Article

The Alexandroff-Urysohn Square and the Fixed Point Property

T. H. Foregger,1 C. L. Hagopian,2 and M. M. Marsh2

1Alcatel-Lucent, Murray Hill, NJ 07974, USA
2Department of Mathematics, California State University, Sacramento, CA 95819, USA

Received 9 June 2009; Accepted 17 September 2009

Academic Editor: Robert Brown

Copyright © 2009 T. H. Foregger et al. 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

Every continuous function of the Alexandroff-Urysohn Square into itself has a fixed point. This follows from G. S. Young's general theorem (1946) that establishes the fixed-point property for every arcwise connected Hausdorff space in which each monotone increasing sequence of arcs is contained in an arc. Here we give a short proof based on the structure of the Alexandroff-Urysohn Square.