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

## TL;DR

This paper investigates the size and structure of transitive quasi-uniformities within a quasi-proximity class, establishing conditions for their cardinality and uniqueness.

## Contribution

It proves the lower bound on the cardinality of transitive quasi-uniformities and characterizes their transitive elements under certain conditions.

## Key findings

- Cardinality of transitive quasi-uniformities is at least $2^{2^{eth_0}}$ when multiple exist.
- Characterization of transitive elements of $ ext{pi}( extdelta)$ when ${ extcal V}_{ extdelta}$ is transitive.
- Conditions for the uniqueness of transitive quasi-uniformity in $ ext{pi}( extdelta)$.

## 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 $\pi(\delta)$.