Proximity relations in transformation groups

Abstract

Introduction. Let (X,T) be a transformation group whose phase space X is a uniform space. Most of the results are stated for compact X. In this paper, four proximity relations, L, M, P, and Q, in X are defined and some of their properties studied. The relations P and Q were first defined and studied by Ellis and Gottschalk [5]. As a general reference for the notions occurring here, consult [8]. In Theorem 1, L is shown to be an invariant equivalence relation in X, and in Theorem 3, L is characterized as the union of all orbit closures under (X x X, T) which are contained in P. Theorem 3 also shows that if P is closed, then P = L and P is a closed invariant equivalence relation in X. Theorem 4 establishes the productivity of the relations Land M, and in Theorem 7, it is shown that (X,T) is coterminous, i.e., P = Q = L= M, iff P = Q. Theorems 9 through 13 describe L, M, P, and Q under various hypotheses such as (X,T) distal; (X,T) minimal and T abelian; (X, T) regionally mixing; and the like. Theorem 14 is an application of the general theory to obtain a characterization of (X,T) being uniformly equicontinuous, and as such represents a strengthening of a theorem by John D. Baum [2]. The author is indebted to Professor W. H. Gottschalk for his invaluable suggestions, mathematical stimulation, and sustained interest. STANDING NOTATION. Let (X,T) be a transformation group where X is always a uniform space. Let / be the class of all syndetic subsets of T, let XV be the class or all compact subsets of T, let OW be the uniformity of X, and for each x E X let