Functional Uniformities
نویسندگان
چکیده
This result suggests that every uniformity on X can be defined by means of a family of sets of real bounded continuous functions, and that important uniformities on X (such as precompact [l ] ones) are definable by means of "nice" sets of such functions. These and related problems are investigated here. A family J of sets of real functions will be said to separate a topological space X if for each PEX and neighborhood NiP) there exists an FEJ and a 5>0 such that if |/(Q)-/(P)| 0, let Va = {P, QEX\ |/(P) -f(Q) \ <8 for all fEF}. Then these symmetric "entourages" Va satisfy the axioms for a uniformity on X. First, naF„=A, the diagonal in X2; for if (P, Q) E Va for every a, it follows that /(P) =/(Q for every
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