On the closure of the diagonal of a $T_1$-space

TL;DR

This paper characterizes which equivalence relations on infinite sets can be realized as the closure of the diagonal in a T_1-topological space, providing insights into the structure of such closures.

Contribution

It offers a characterization of equivalence relations that are closures of the diagonal in T_1 spaces, linking topological properties with algebraic relations.

Findings

Identifies conditions for equivalence relations to be diagonal closures in T_1 spaces

Provides a complete characterization for infinite sets

Connects topological closure properties with algebraic relations

Abstract

Let X be a topological space. The closure of \Delta = {(x, x) : x \in X} in X \times X is a symmetric relation on X. We characterise those equivalence relations on an infinite set that arise as the closure of the diagonal with respect to a T_1-topology.