A direct proof of Tychonoff's theorem

TL;DR

This paper provides a straightforward and accessible proof of Tychonoff's theorem using the open cover definition of compactness, avoiding complex machinery like ultrafilters or nets.

Contribution

It introduces a direct and simple proof of Tychonoff's theorem based solely on the open cover definition of compactness, simplifying understanding.

Findings

Proof avoids ultrafilters, nets, and maximal families.

Demonstrates a direct proof from open cover definition.

Simplifies the conceptual understanding of Tychonoff's theorem.

Abstract

Proofs of Tychonoff's theorem often seem to require a bit of magic. Machinery such as ultrafilters, nets or maximal families with the finite intersection property are employed to give proofs that can be very neat, but not the kind of thing that one would naturally think of when presented with the problem (given a background in standard open set topology). Here we present a direct and pretty simple proof of Tychonoff's theorem, straight from the open cover definition of compactness.