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By Jack Feuillet

ISBN-10: 3895860107

ISBN-13: 9783895860102

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Suppose that (X,T) is hereditarily com­ pact and let 6 be a quasi-proximity that is compatible with T. ôp C Ô. Suppose that A (fp B. 43 that A (f B. (a) ^ (b) : Let U be a quasi-uniformity compatible with T. is hereditarily compact, = 6p so that P c H. As (X,T) Since T is finite. Up e P. Furthermore every neighbornet of X contains U^; hence Cl c p. 8 that every interior-preserving open collection is finite. (c) => (d) : It is well known that a topological space is hereditarily compact if and only if every strictly increasing sequence of open sets is finite.

Since UA is a subbase for T, there ... ,e UA such that x e Thus T C Т(0д) so that Now let is an interior-preserving open collection so Thus e A such that [U^ (x)] c G. e C^. c G. Then For each i with I < e are A^, i

N such that Since D and F n F = 0 - a contradiction. N set {y e X X x|d(y,K) < 1/n}. Evidently For each n e W set = {(x,x)|x e I}. e Tí} is a base for the fine uniformity, which consists of all neighborhoods of the diagonal. ) Since X is metrizable, it is an immediate consequence of Proposition 2,25 that the fine uniformity is equinormal. 34 This implication is evident. ■ We now characterize those regular uniformity has a countable base. 33(b), to which it should be compared. PROPOSITION. Let (X,T) be a regular T^ space.

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Bulgare by Jack Feuillet

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