Fixed-Point Theorems for Arcwise Connected Continua

Abstract

recently proved in [4] a fixed-point theorem for certain arcwise connected spaces that generalizes a theorem of mine-Theorem 2, below-and Borsuk's theorem [l] that an arcwise connected hereditarily unicoherent metric curve has the fixed-point property.His argument provides a proof of my result, but not of Borsuk's.That Borsuk's class of continua is contained in his follows from Borsuk's result only.In this note I give a new sufficient condition for the fixed-point property that implies Borsuk's result, and that follows from my theorem and so from Ward's.I also give an example of an arcwise connected continuum that contains no simple closed curve but that does not have the fixed-point property, and prove a fixed-point theorem for a quite special class of contractible continua. Theorem1. Let M be an arcwise connected compact Hausdorff space that does not have the fixed-point property.Then M contains either (1) a continuum Nifor which there is a map f: N1-+S1 which is onto and such that no closed proper subset of Nt is mapped by f onto S1, and which is such that at most one point-inverse is nondegenerate, that one being connected; or (2) a continuum N2 that contains a subset R that is the oneto-one continuous image of a half-open interval and that is dense in N2, but that has no interior relative to N2; or (3) a continuum Ns that is the union of a set R that is the continuous one-to-one image of a half-open interval, and a continuum B, and for which there is a map f: N3->K, K being the union of the circles x2+y2 = (2/«)y, w = l, 2, 3, • • -, such that f is one-to-one on N3 -B, such that f(B) = (0, 0), and such that no closed proper subset of Ns is mapped by f onto K.We will see that this is a consequence of an earlier fixed-point theorem of the author's, proved in [5, p. 493]: Theorem 2. Let M be an arcwise connected Hausdorff space which is such that every monotone increasing sequence of arcs is contained in an arc.Then M has the fixed-point property.Note that compactness is not required in Theorem 2.