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• W2065519899 abstract "We show that an example of Burke and van Douwen has no complete quasi-uniformity. Moreover, we show that it is almost finitely-fully normal but not almost No-fully normal. 0. Introduction. Every topological space admits a quasi-uniformity. The problem whether every topological space admits a complete quasi-uniformity is considered in [3, Problem C], where an example is given of a T,-space that admits a complete, but no convergence complete, quasi-uniformity. In this note we show that a locally compact separable normal M-space of D. K. Burke and E. K. van Douwen admits no complete quasi-uniformity, thereby answering an old question in the theory of quasi-uniform spaces. Moreover, we show that this space is an almost finitely-fully normal countably paracompact space that is not almost t 0-fully normal. It is interesting to compare these results with the recent results of K. P. Hart [4] that M. E. Rudin's Dowker space is both orthocompact and finitely-fully normal; for it follows readily from Hart's results that, while Rudin's space is not almost t 0-fully normal, it does admit a complete quasi-uniformity. 1. Definitions and a lemma. A quasi-uniformity on a set X is a filter 1, on X X X such that (a) each member of 1t is a reflexive relation on X, and (b) if U E qI then V o V C U for some V E 9t. The pair (X, 91) is called a quasi-uniform space. A filter F on (X, 91) is a Cauchy filter provided that for each U E q9 there exists p E X so that U(p) E C, and (X, 91) is said to be complete provided that every Cauchy filter has a cluster point. The topology T(qt) = {G C X: for each x E G there is U E Qt with U(x) C G} is called the topology induced by 91. A topological space (X, T) admits 91 provided that T is the topology induced by 9t. Let (X, T) be a topological space and let 63 be the collection of reflexive transitive relations V on X for which V(x) E T for all x E X. Then 6J3 is a filterbase for a quasi-uniformity 91. Moreover, using the observation of W. J. Pervin [8] that for each open set G, G X G U (XG) X X E 91, we see that (X, T) admits 91W. It is known that this quasi-uniformity 91 is complete if and only if every ultrafilter on X without a cluster point has a closure-preserving subcollection without a cluster point [3, p. 59]. Consequently, a Received by the editors February 14, 1983 and, in revised form, June 28, 1983. 1980 Mathematics Subject Classification. Primary 54E15, 54D20. Keiy words and phrases. Almost 0-fully normal, almost finitely-fully normal, complete quasi-uniformity. 'While working on this paper, the first author was supported by the Schweizerischer Nationalfonds. D1 984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page" @default.
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• W2065519899 title "A topological space without a complete quasi-uniformity" @default.
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• W2065519899 doi "https://doi.org/10.1090/s0002-9939-1984-0733415-3" @default.
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