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Extra info for Quasi-Uniform Spaces
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.