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.

So, first, here is the key definition.

Definition  Let \(S\) be a subset of a topological space \(X\).  A point \(x\in X\) is a perfect limit point 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 perfect limit point.

I’ll give a proof in a later post.  Meanwhile, let’s assume it and prove Tychonoff’s theorem.  Suppose that \(X\) is a product space \(\prod_{\alpha\in{\mathcal A}} X_\alpha\) where all \(X_\alpha\) are compact.  Let \(S\) be an infinite subset of \(X\).

We consider “partially defined” perfect limit points. A PPLP (for short) is a subset of \(X\) given by assigning definite values to some of the coordinates, i.e. an intersection of \(\pi_\alpha^{-1}(x_\alpha)\) for \(\alpha\) belonging to some (possibly empty) subset of \(\mathcal A\), which has the property that any neighborhood of it (in the product topology) contains \(\text{card}(S)\) points of \(S\).  For example, \(X\) itself is a PPLP.

The collection of PPLPs is partially ordered by reverse inclusion.  In this partially ordered set, every chain has an upper bound given by its intersection – that this intersection is a PPLP follows from the definition of neighborhoods in the product topology.  Thus, Zorn’e lemma tells us that there is a maximal PPLP.  I assert that it is a singleton, i.e., a perfect limit point of \(S\).

Suppose not.  Then the maximal PPLP is of the form \(X_{\alpha_0}\times Y\) where \(Y\) is some subset of \(\prod_{\alpha\neq\alpha_0} X_\alpha\).  I claim that some point \(x_0\) of \(X_{\alpha_0}\) has the property that, for every neighborhood \(U\) of \(x_0\) in \(X_{\alpha_0}\) and every neighborhood \(V\) of \(Y\) in \(\prod_{\alpha\neq\alpha_0} X_\alpha\), we have \(\text{card}((U\times V)\cap S) = \text{card}(S)\).  Indeed, if not, then \(X_{\alpha_0}\) would be covered by neighborhoods \(U\) having \(\text{card}((U\times V)\cap S) < \text{card}(S)\) for every \(V\).  Such a cover would have a finite subcover, by compactness; and this would contradict the fact that \(X_{\alpha_0}\times Y\) is a PPLP (a finite union of sets of cardinality less than \(\text{card}(S)\) is also a set of cardinality less than \(\text{card}(S)\)).

But now it follows that \(\{x_0\}\times Y\) is a PPLP also, and this contradicts maximality. The proof is finished.

Reference

Wright, David G. “Tychonoff’s Theorem.” Proceedings of the American Mathematical Society 120, no. 3 (1994): 985–87.

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