By S. Zaidman

ISBN-10: 0273086618

ISBN-13: 9780273086611

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**Extra info for Almost-periodic functions in abstract spaces**

**Example text**

Let n be a positive integer and n) Ix EX}. Then u {B n} is a basis for the topology generated put B n = {S(x, 2-n by d. Let U be an open covering of X and pick U E U such that x E U. Then there is a positive integer m such that Sex, 2- mm ) c U. Now so that {B n} is locally starring for all open coverings of X. Conversely, assume X X is T 1 and there exists a sequence {V n} of open coverings of X that is locally starring for all open coverings of X. It will first be shown that X is paracompact.

2(a». This shows X is metrizable. Next it will be shown that the topological product X of a sequence {X {Xn} n } of metric spaces is metrizable. For each positive integer n let d n denote the metric for X n • First we want to show that for each n it is possible to construct a new Ynn in X n. 12) for each pair of points XXnn and Yn in X n. Thatlnn is a metric for X Xnn that generates Xnn is left as an exercise (Exercise 2(b». 2(b)). 13) for any pair of points x = = (x 11,, X 2, . . ) and Y == (y 11,, Y 2, .

To show dEl (H(z» (H(z» and dEl (K(z» (H(z» is open (H(z» = = 0 which is open. Consequently we may ~ 0 then dEl (H(z» note that if z :S: b)/2. Since assume z > O. Let p E dEl (H(z». Then dE(P) = b 0 < z. Put ££ = (z - 0)/2. dEep) E. Let U = dE(P) = d(p, E) = b 0 there exists a point q E E with d(p,q) < 0b + £. S(P,£). Then x E U implies that S(P,E). = d(x, d(x,E):S:d(x,q):s: d(x,p)+d(P,q) dE(x) = E) ~ d(x, q) ~ d(x,p) + d(p, q) < E£+0+£ +b+E = z. dEl (H(z» so that U c dEl (H(z». Therefore dEl (H(z» is open.

### Almost-periodic functions in abstract spaces by S. Zaidman

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