Tag Archives: axiom of choice

Tychonoff’s theorem II

In my earlier post on Tychonoff’s theorem, I talked about the original proof, based on the following characterization of compactness which is due to Kuratowski.

Definition  Let \(S\) be a subset of a topological space \(X\).  A point \(x\in X\) is a point of perfection of \(S\) if, for every neighborhood \(U\) of \(x\), the set \(U\cap S\) has the same cardinality as \(S\).

Lemma (Kuratowski) A topological space \(X\) is compact if and only if every infinite subset has a point of perfection.

Now I will give the proof of this characterization (again following Wright).

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Tychonoff’s theorem

Tychonoff’s theorem (an arbitrary product of compact sets is compact) is one of the high points of any general topology course.  When I’ve taught this in recent years, I’ve usually given the proof using universal nets, which I think is due to Kelley.

Recently though I read a very nice paper by Wright  which reproduces, and then modifies, Tychonoff’s original proof (otherwise inaccessible to me because of my lack of German).  I thought the original proof was really elegant and though I would try to give an exposition.

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