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

Oleksiy Dovgoshey, Ruslan Shanin

TL;DR

The paper characterizes when a prescribed family of singleton closures arises from a topology on a set X that satisfies the $T_0$ separation property, showing this is equivalent to realizability within an $T_0$-Alexandroff topology. It proves a key equivalence (Theorem t3.1) between extending X -> 2^X to a closure operator in a $T_0$-space and the same extension existing in a $T_0$-Alexandroff topology, and establishes that every $T_0$-Alexandroff space is quasi-metrizable by an equidistant quasi-metric (Theorem t3.4). The results bridge closure operators, Alexandroff topologies, and equidistant quasi-metric structures, with implications for the uniqueness of Alexandroff representations and related quotient classifications of $T_0$-topologies. Overall, the work provides concrete criteria and constructive methods linking single-point closures to metric-like topological representations in $T_0$-settings.

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 $(X, τ)$ are described. It is proved that every $T_0$-Alexandroff space is quasi-metrizable by some equidistant quasi-metric.

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

TL;DR

The paper characterizes when a prescribed family of singleton closures arises from a topology on a set X that satisfies the separation property, showing this is equivalent to realizability within an -Alexandroff topology. It proves a key equivalence (Theorem t3.1) between extending X -> 2^X to a closure operator in a -space and the same extension existing in a -Alexandroff topology, and establishes that every -Alexandroff space is quasi-metrizable by an equidistant quasi-metric (Theorem t3.4). The results bridge closure operators, Alexandroff topologies, and equidistant quasi-metric structures, with implications for the uniqueness of Alexandroff representations and related quotient classifications of -topologies. Overall, the work provides concrete criteria and constructive methods linking single-point closures to metric-like topological representations in -settings.

Abstract

Let be a set and be a set of all subsets of . The necessary and sufficient conditions under which a mapping is a closure of one-point sets in some -space are described. It is proved that every -Alexandroff space is quasi-metrizable by some equidistant quasi-metric.

Paper Structure

This paper contains 3 sections, 12 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The closure of points in $T_0$-spaces and Alexandroff $T_0$-spaces

Key Result

Theorem 2.1

Let $X$ be a set and let a mapping $\operatorname{cl} \colon 2^X \to 2^X$ satisfy the conditions: for all $A$, $B \subseteq X$. Then there exists a topology $\tau$ on $X$ such that for each $A \subseteq X$.

Theorems & Definitions (28)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Corollary 2.9
  • ...and 18 more