A fixed point theorem

Abstract

Suppose that space is metric. A chain is a finite collection of open sets d1, d2, * * * , dn such that di intersects dj if and only if I i-jj I 1. If the elements of a chain are of diameter less than a positive number e, that chain is said to be an e-chain. A compact continuum is said to be chainable if for each positive number e, there is an e-chain covering it. R. H. Bing has called [1] such continua snake-like. In 1951 0. H. Hamilton showed [4] that every compact chainable continuum has the fixed point property; i.e., that if f is a continuous mapping of such a continuum M into itself, then some point of M is its own image underf. In the present paper it is shown that the Cartesian product of finitely many compact chainable continua has the fixed point property. Since arcs are compact chainable continua, this is a generalization of the Brouwer fixed point theorem. Two other examples of compact chainable continua are the closure of the graph of sin (1/x), 0 <x _1, and the pseudo-arc. Other examples are given in [1]. After reading the original manuscript of this paper, A. D. Wallace raised the question as to whether the Cartesian product of finitely many compact chainable continua is a quasi-complex (p. 323 of [6]). Rather surprisingly, the answer to this question is in the affirmative. A proof of this theorem is also given here. Since the Cartesian product of finitely many compact chainable continua is zero-cyclic, the fact that such products have the fixed point property is a special case of the Lefschetz fixed point theorem for zero-cyclic quasi-complexes. Since the author's original argument is of a very different nature, it is also given. Let En denote Euclidean n-space and Rn the set of all points of En whose distance from the origin is not greater than one. Let Sn-1 denote the set of all points of En whose distance from the origin is one. Let I denote the set of all points on the x-axis having abscissa x such that 0< x ?1, and let In denote the set of all points of En each of whose n coordinates xi satisfies 0 -xi_ 1. If P is a point of En having coordinates (x1, x2, * * *, xn) and t is a real number, by tP is meant the point of Es having coordinates (tx1, tx2, * . , txn).