A regular space without a uniformly regular quasi-uniformity

Abstract

Abstract.We point out that a construction due to J. Vaughan yields an example of a regular T~-space that does not admit a uniformly regular quasi-uniformity.In [4] P. FLETCHER, J. HEJCMAN and W. HUNSAKER ask whether each regular Tl-space admits a quiet ([2]) quasi-uniformity. In this note we answer this question in the negative. Since each quiet quasi-uniformity is uniformly regular ([5, Proposition 1.2], [2, Proposition 5]), it suffices to exhibit a regular T~-space that does not admit a uniformly regular quasi-uniformity. Let us recall that a quasi-uniformity q/on a set X is called uniformly regular ([5], [1, p. 141]) provided that for each UE q/ there is a Ve~ such that V(x)~_ U(x) whenever xeX. Since each countably compact first countable Tz-space is regular, our result is an immediate consequence of the following two facts. Proposition 1. [7, Remark] There is a countably compact first countable T2-space that is not completely regular.Proposition 2. A eountably compact sequential T2-space X that admits a uniformly regular quasi-uniformity ~ is completely regular.