A characterization of the topological spaces that possess a bicomplete fine quasiuniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable 7^-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T t -spaces that do not admit a bicomplete quasi-uniformity. We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the

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... monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober. available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0305004100069644 Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:33:47, subject to the Cambridge Core terms of use, are W e "U with W 2 £ V and A e & with AxA^Wf] W' 1 . Since p e cl gm A, we have an aeA f] W(p). Then, for any xeA, we get (p,x)e W 2 , whence xe V(p), so that A c V(p), and V(p)e&. (b) See [22], lemma 1. LEMMA 2 . Let X be a topological space and let ~V be a compatible quasi-uniformity on X that is finer than the Pervin quasi-uniformity 0> of X. (a) Let ?F be a ~V*-Cauchy filter on X, and let x eX. Then !F converges to x with respect to the topology S~^f"*) if and only ifx is a 3T{'f) -cluster point of 2F and cl^-(y -) {x} belongs to &. (b) The topologies ST{-r*) and y(@>*) are equal. (c) If X is a T D -space, the quasi-uniformity "V is bicomplete and ~W is an arbitrary {possibly not compatible) quasi-uniformity finer than "V onX, then iV is bicomplete, too.