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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article The Alexandroff-Urysohn Square and the Fixed Point Property | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 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 1 Alcatel-Lucent Murray Hill NJ 07974 USA 2 Department of Mathematics California State University Sacramento CA 95819 USA Correspondence should be addressed to M. M. Marsh mmarsh@csus.edu Received 9 June 2009 Accepted 17 September 2009 Recommended by Robert Brown 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. 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. Alexandroff and Urysohn 1 in Memoire sur les espaces topologiques compacts defined a variety of important examples in general topology. The final manuscript for this classical paper was prepared in 1923 by Alexandroff shortly after the death of Urysohn. On 1 page 15 Alexandroff denoted a certain space by u1. While Steen and Seebach in Counterexamples in Topology 2 Example 101 refer to this space as the Alexandroff Square we concur with Cameron 3 pages 791-792 who attributes it to Urysohn. Hence we refer to U1 as the Alexandroff-Urysohn Square and for convenience denote it by X T . The following definition of X T is given by Steen and Seebach 2 Example 101 pages 120-121 . Define X to be the closed unit square 0 1 X 0 1 with the topology T defined by taking as a neighborhood basis of each point s t off the diagonal A x x e X x e 0
