On the cardinality of $\pi(\delta)$

Abstract

We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least $2^{2^{\aleph_0}}$ if there exist at least two transitive quasi-uniformities in the class. The transitive elements of $\pi(\delta)$ are characterized if ${\cal V}_{\delta}$ is transitive, and in this case we give a condition when there exists a unique transitive quasi-uniformity in $\pi(\delta)$.