A fixed point theorem for plane continua

Abstract

In this paper it is proved that every bounded arcwise connected plane continuum which does not separate the plane has the fixed point property. A set X is said to have the fixed point property if each continuous function ƒ on X into itself leaves some point fixed (that is, there is a point x belonging to X such that f(x)=x). The problem Must a bounded plane continuum which does not separate the plane have the fixed point has motivated a great deal of research in plane topology. K. Borsuk in 1932 proved that every Peano continuum which lies in the plane and does not separate the plane has the fixed point property [2]. Since that time, other general conditions have been found which insure that a plane continuum has this property. In 1967, H. Bell proved that every bounded plane continuum which does not separate the plane and has a hereditarily decomposable boundary has the fixed point property [ l ] . The following question is still outstanding. If a bounded plane continuum is arcwise connected and does not separate the plane, then must it have the fixed point property? Here an affirmative answer is given to this question by proving the following theorem. If M is a bounded arcwise connected plane continuum which does not have infinitely many complementary domains, then the boundary of M does not contain an indecomposable continuum. Throughout this paper 5 is the set of points of a simple closed surface (that is, a 2-sphere). The proof of the following theorem is based on techniques which are closely related to the folded complementary domain concept defined by F. Burton Jones [3, p. 173]. THEOREM 1. Suppose M is a continuum in Sy S—M does not have AMS 1970 subject classifications. Primary 54H25, 57A05, 47H10, 55C20; Secondary 54F22.