Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 310832, 3 pages
doi:10.1155/2009/310832
The Alexandroff-Urysohn square and the fixed point property
T.H. Foregger1
, C.L. Hagopian2
and M.M. Marsh2
1Alcatel-Lucent, Murray Hill, NJ 07974, USA
2Department of Mathematics, California State University, Sacramento, CA 95819, USA
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.