A non-completely regular quiet quasi-uniformity

Abstract

Abstract.A complete quiet quasi-uniformity is constructed on A. Mysior's regular but not completely regular space. This answers a question raised by D. Doitchinov at the Prague conference on categorial topology in 1988.It is well known that the classical theory of completeness for uniform spaces cannot be extended satisfactorily to the class of all quasi-uniform spaces. In [ 1 ] D. Doitchinov introduced the class of quiet quasi-uniform spaces and gave a theory of completion for them. His theory of completion is as well behaved as the classical theory of completion for uniform spaces, and his class of quiet quasi-uniform spaces includes all uniform spaces and many of the most interesting quasi-uniform spaces. He developed topological properties of these spaces in [2], where he established that every quiet quasi-uniform space is a regular Hausdorff space and asked whether each quiet space is completely regular. This question was motivated by his previous result that every balanced quasi-metric space is a Tychonoff space [3, Corollary 3]. In this note we construct a quiet transitive quasi-uniformity that is complete in the sense of Doitchinov and is compatible with A. Mysior's noncompletely regular space [6].The definitions due to Doitchinov needed in this paper are given below; for further information on quasi-uniform spaces see [4].