Yet another proof of Tychonoff's Theorem

TL;DR

This paper presents two versions of a proof of Tychonoff's Theorem, based on closed projections and a characterization of compactness, including a recent generalization that offers a new perspective.

Contribution

It provides two distinct proofs of Tychonoff's Theorem using closed projection techniques, including a recent generalization that offers a novel approach.

Findings

Two different proofs of Tychonoff's Theorem are presented.

A recent generalization of closed projection results is applied.

The proofs offer new insights into the theorem's foundations.

Abstract

In 1971 I announced what I described as a nice proof of Tychonoff's Theorem, an immediate corollary of a result concerning closed projections combined with Mrowka's characterization of compactness: a space X is compact if and only if for each space Y the projection from X x Y to Y is closed. I described the proof as to appear but to date it has not. In 2019 I published a generalization of the stronger closed projection result which yields a different look to the proof. Both versions are presented here.