An order relation among topological spaces

Abstract

In this paper we define a partial ordering on a class of topological spaces as follows('): For two spaces X and Y in the class, we will say X Y which is an onto a-map. (Recall that f is an a-map if there is an open cover ,B of Y such that f-'(13) refines a.) It is easy to check that the relation < is reflexive and transitive. In [7], Kuratowski proved that the closed n-dimensional ball is not _ the n-sphere. More recently, Ulam [8] raised the question whether the 2-dimensional ball is < the 2-dimensional torus. Ulam's question has stimulated interest in the ordering itself, and a result of T. Ganea [5] states, roughly speaking, that if X < Y and Y is a compact n-dimensional manifold and X is an absolute neighborhood retract, then X has the same homotopy type as an n-dimensional manifold. This result then gives Ulam's question a negative answer(2), and suggests studying properties of a space Y which are inherited by a space X satisfying X _ Y. The purpose of this paper is to present certain properties which are inherited, in the above sense. Also, comparison is made between this ordering and the ordering given by dimension (Theorem 1), and (for the class of absolute neighborhood retracts) the ordering given by a-domination for every cover a (Theorem 2). Since the property of possessing a fixed point under every continuous function will be shown to be inherited, another question of Ulam's is answered here [8, Chapter IV, Problem 12].