Almost Periodicity, Equi-continuity and Total Boundedness

Let X be a uniform space; that is to say, let X be a space provided with a system of indexed neighborhoods Ua(x) (x(EX, a=index), subject to the conditions (A. Weil) : (1) If xÇzX and if a is an index, then xÇzUa(x)\ (2) If a and /? are indices, then there exists an index y such that xÇzX implies U7(x)ÇZUa(x)r\Up(x); (3) If a is an index, then there exists an index /? such that x, y, zÇzX with x, yÇzUp(z) implies xÇzUa(y)Let T be a topological group with identity <r and let ƒ be a transformation of XXT into X. We agree to write f'(x) or fx(t) in place of f(x, t) (xÇzX, t£.T), whenever we wish. Furthermore, let ƒ define a transformation group; that is to say, suppose f*(x)=x and ff(x)=f(x) (xÇzX; t, S £ J T ) . We impose continuity conditions on ƒ as the needs arise. A subset E of T is said to be relatively dense provided there exists a compact set A in T such that each left translate of A intersects E. A point x of X is called almost periodic provided that if U is a neighborhood of x, then there exists a relatively dense set E in T for which f(x, £ ) C £f. We observe that the notion of almost periodic point depends on the topology in T, the strongest type of almost periodicity occurring when T is provided with the discrete topology. I t is easily proved tha t a set E in T is relatively dense if and only if there exists a compact set B in T such that T=EB. The set f(x, T) is called the orbit of the point x.