On the closure of one point sets in \(T_0\)-spaces

TL;DR

This paper characterizes when a set-valued mapping can be realized as the closure of singletons in a T0-space and proves that all T0-Alexandroff spaces are quasi-metrizable by an equidistant quasi-metric.

Contribution

It provides necessary and sufficient conditions for a set-valued map to be a closure operator of one-point sets in T0-spaces and establishes quasi-metrizability of T0-Alexandroff spaces.

Findings

Characterization of closure of one-point sets in T0-spaces.

Conditions for a set-valued map to be a closure operator.

T0-Alexandroff spaces are quasi-metrizable by equidistant quasi-metrics.

Abstract

Let $X$ be a set and $2^X$ be a set of all subsets of $X$. The necessary and sufficient conditions under which a mapping $X \to 2^X$ is a closure of one-point sets in some $T_0$-space $(X, \tau)$ are described. It is proved that every $T_0$-Alexandroff space is quasi-metrizable by some equidistant quasi-metric.