𝑚-bounded uniformities between two given uniformities

Abstract

Introduction. This paper is an attempt to gain some information about the lattice of the uniformities which can be defined on a set X. Our results are stated in terms of the m-boundedness of uniformities, where m is an infinite cardinal. We define a pseudo-metric p on X to be m-bounded if for e > 0, X can be written as a union of fewer than m sets, each of p-diameter not exceeding E. If P is a family of m-bounded pseudo-metrics on X we say that the associated uniformity &(P) is m-bounded. We call &(P) strictly m-bounded if m is its least (infinite) bound. Roughly speaking, we found that if two uniformities 9 and 91 have different strict bounds, with 90c l, then there are at least c uniformities between them. (Here c is the cardinality of the continuum.) The lower bound on the number of uniformities in-between can be improved as the strict bound on 91 increases. Moreover, these uniformities can be chosen to be strictly m-bounded, provided m exceeds the bound on 90 (but not the bound on Sb). These results throw some light on the structure of a proximity class. For example, if a proximity class has a non-totally-bounded member then it must contain at least c uniformities. If a proximity class has a member which is strictly m-bounded then for every cardinal n between No and m the class has a strictly n-bounded member (in fact, quite a few). Thus, no gaps are allowed as far as n-boundedness is concerned. Finally, no proximity class has a minimal m-bounded uniformity, for mr>NO. We would like to express our thanks to the referee for simplifying our proofs and our results, and indeed for strengthening our results. From his improvements we were able to obtain further extensions. Theorem 2.1 is in fact a combined effort.