proof of Tychonoff’s theorem

This is a proof in of nets. Recall the following facts:

1 - A net (xα)α𝒜 in iIXi convergesPlanetmathPlanetmath to xiIXi if and only if each coordinate (xαi)α𝒜 converges to xiXi

2 - A topological spaceMathworldPlanetmath X is compactPlanetmathPlanetmath if and only if every net in X has a convergent subnet.

3 - Every net has a universalPlanetmathPlanetmath subnet.

4 - A universal net ( (xα)α𝒜 in a compact space X is convergent. (see this entry (

We now prove TychonoffPlanetmathPlanetmath’s theorem.

Proof (Tychonoff’s theorem) : Let (xα)α𝒜 be a net in iIXi.

Using Lemma 3 we can find a subnet (yβ)β of (xα)α𝒜.

It is easily seen that each coordinate net (yβi)β is a net in Xi.

Using Lemma 4 we see that each coordinate net converges, because Xi is compact.

Using Lemma 1 we see that the whole net (yβ)β converges in iIXi.

We conclude that every net in iIXi has a convergent subnet, so, by Lemma 2, iIXi must be compact.

Title proof of Tychonoff’s theorem
Canonical name ProofOfTychonoffsTheorem
Date of creation 2013-03-22 17:25:24
Last modified on 2013-03-22 17:25:24
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 8
Author asteroid (17536)
Entry type Proof
Classification msc 54D30