Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Combinatorial properties of uniformities Jan Pelant, Prague (Czechoslovakia) In [S^» A.H.Stone raised a question of whether each uniform space has a basis consisting of locally finite covers (recall the A.Stone theorem asserting that each metric space is paracompact).It is shown easily in jjQ that the existence of a basis consisting of locally finite covers is equivalent to the existence of a basis consisting of point-finite covers. Stone's problem is restated in [l^[ and other related problems are pointed out (e.g. the problem of when the Ginsburg-Isbell derivative forms a uniformity, see [pj,[PPV]). The negative answer to Stone's problem was given independently by E.SSepin and myself in 1975. Hence the class of all spaces with a point-finite basis forms a "nice* 1 proper epireflective subcategory of UNIF. However, it appears that even spaces having point-finite bases are very wild and that perhaps the best uniform spaces are those having bases consisting of (T-disjoint covers. (A Gf-disjoint basis implies the existence of a point-finite base (see e.g. Q®3't^ll ' but the converse is not true, (see [Pgl^* This paper illustrates the use of "combinatorial" (or discrete) reasoning, as opposed to M continuous" reasoning, in the theory of uniform spaces. This approach seems particularly applicable to problems dealing with covering properties of uniformities. We are going to estimate point character of some uniform spaces. Finally, we show that the properties of cardinal reflections in UNIF depends on set-theoretical assumptions.
