An Equicontinuity Condition in Topological Dynamics

The reference for definitions both in the above theorem and in what follows is Topological dynamics by W. H. Gottschalk and G. A. Hedlund, Providence, 1955. It is tacitly assumed both in the statement of the theorem and in the remainder of the paper that the uniformity of X induces the (Hausdorff) topology. We observe that it is obvious that (II) implies (I); the remainder of the paper is devoted to proving the converse statement. For each of the following lemmas we take the following three statements as hypotheses: (1) X is a compact Hausdorff space, (2) For each index a of X there exist an index ß of X and a replete semigroup P^CTsuch that (x, y)£/3 implies (xp, yp)Ga for each p(EPaß, (3) T is an abelian topological group. Full use of all these hypotheses is not made in every lemma. Throughout the paper Paß will be used to denote a replete semigroup with the property mentioned in (2) above with respect to the indices a and ß.