On the cardinality of compatible quasi-uniformities

In this paper we begin to study some cardinal functions related to quasi-uniformities. We show that there exist either a unique or at least 22ℵ0 compatible quasi-uniformities on a topological space. The cardinality of the set of non-transitive compatible quasi-uniformities is examined and it is shown that for an infinite T2 space there are at least 22ℵ0 such quasi-uniformities. We also derive that the cardinality of the set of the compatible quasi-uniformities is 22ℵ0 if X=R.