The topological fixed point property - an elementary continuum-theoretic approach

A set contained in a topological space has the topological fixed point property if every continuous mapping of the set into itself leaves some point fixed. In 1969, R. H. Bing published his article The Elusive Fixed Point Property, posing twelve intriguing and difficult problems, which exerted a great influence on the study of the fixed point property. We now present a survey article intended for a broad audience that reports on this area of fixed point theory. The exposition is also intended to give an introduction to the current study of the fixed point property from the viewpoint of an elementary continuum theory. This article is an expanded version of an invited lecture given by the author at the International Conference on Fixed Point Theory and its Applications in Memory of Jim Dugundji (1919–1985), Będlewo, Poland, August 1st to 5th, 2005. The author wishes to thank Kazuhiro Kawamura and Mirosław Sobolewski for several suggestions that led to the improvement of this article. The author’s special thanks are due to Charles L. Hagopian for his careful reading the penultimate typescript and supplying valuable remarks and comments that corrected the article. Special thanks are also due to the referee for valuable suggestions.