Quasi-Uniform Spaces by Fletcher

By Fletcher

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Moreover every transitive neighbornet of S is the restriction to S of transitive neighbornet of X. P roof. Vu Let V be a normal neighbornet of S. (X - S X X) and if S is closed, set If S is open, set = V и (X x X - S ) . the required neighbornet of X and it is easily seen that = Then is is transitive whenever V is transitive. ■ COROLLARY. Let X be a topological space and let a subset S of X be the intersection of an open set and a closed set. Then FIME^|S x S = and FT^|S X S = FTg. 20 We close this section with a technical theorem, which shows in par­ ticular that the restriction of the fine quasi-uniformity on IR to the subset Q is not the fine quasi-uniformity on Q.

As T(A,B) = X - A x X u Hence I/ is finer than The notation ■ denotes the totally bounded U is standard and, as we will not use ш to denote a quasi-proximity, no ambiguity will arise. 33 that U is the finest totally bounded quasi-uniformity on X that is coarser than Li and that two quasi-uniformities LI and I/ are qp-equivalent if and only if Ü = I/ . 38 We consider the family of all quasi-proximities on a set X to be partially ordered by than P (and p is reverse set inclusion and we say that ô is coarser f in e r than ô) provided p c ô.

P roo f, As in the previous proposition, let p denote the coarsest quasi-proximity compatible with is uniformly continuous. 21, f : By the previous proposition, (Y,l/) <= U and so the result follows. 45 naturally suggest two questions: Which topological spaces admit only one quasi-proximity and which spaces admit a coarsest quasi-proximity? We do not know the com­ plete answer to either of these questions and we postpone considering the first until the next chapter. The following theorem provides an answer to the second question for Tychonoff spaces.

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