Ring-theoretic approaches to point-set topology

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.