Ring-theoretic approaches to point-set topology

TL;DR

This paper explores the connection between ring theory and topology, showing that certain topological properties can be characterized by maximal ideals in the power set ring, and provides ring-theoretic proofs of classical theorems.

Contribution

It introduces novel ring-theoretic characterizations of compactness, Tychonoff, Alexander subbase, and profinite spaces, offering new proofs for these fundamental results.

Findings

A topological space is compact iff every maximal ideal of its power set ring converges to exactly one point.

Ring-theoretic proofs of Tychonoff and Alexander subbase theorems are provided.

Profinite spaces are characterized as compact and totally disconnected spaces using ring theory.

Abstract

In this paper, it is shown that a topological space $X$ is compact iff every maximal ideal of the power set ring $\mathcal{P}(X)$ converges to exactly one point of $X$. Then as an application, simple and ring-theoretic proofs are provided for the Tychonoff theorem and Alexander subbase theorem. As another result in this spirit, a ring-theoretic proof is given to the fact that a topological space is a profinite space iff it is compact and totally disconnected.