By Walter Helbig Gottschalk and Gustav Arnold Hedlund
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PROOF. 17. 20. For the remainder of this section X denotes a uniform space. 21. REMARK. The following statements are pairwise equivalent: (1) T is weakly almost periodic; that is to say, if a is an index of X, then there exist a left syndetic subset A of T and a compact subset C of T such that x E X implies the existence of a subset B of T for which A C BC and xB Cxa. (2) If a is an index of X, then there exists a compact subset K of T such that x E X implies the existence of a subset A of T for which T = AK and xA C xa.
Is recursive at x. , is recursive at x, then S is recursive at x. Suppose Sx is recursive at x. Let U be an open neighborhood of x. , C SM- Let V be a neighborhood of x for which VM C U. , such that xA C V. Now xAM C U. Define B = S n AM. Since A C BM-\ B is a T-admissible subset of T. Also B C Sand xB C U. Thus S is recursive at x. It now follows that if T is recursive at x, then S is recursive at x. The converse is obvious. The proof is completed. 37. DEFINITION. Let T be a topological group.
The expression weakly almost periodic was introduced by Gottschalk . 36) Cf. Gottschalk [2, 6, 8], Erdos and Stone , Gottschalk and Hedlund . 38) The terms replete and extensive, as defined here, were introduced by Gottschalk and Hedlund . If T is either 9 or (ft, a subset A of T is extensive if and only if A contains a sequence marching to + CD and a sequence marching to - CD. 55] The expression almost periodic, as applied to a point, is a generalization of the term recurrent as used by G.