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 ${\mathcal{V}}_{\delta }$ is transitive, and in this case we give a condition when there exists a unique transitive quasi-uniformity in $\pi (\delta )$.