Talk:Tychonoff's theorem

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Where is it written never to use \empty?? Maybe this is true for HTML, where the empty set symbol doesn't fit well within a line of inline type. But, why should it not be used in math/tex mode when on a separate line?? The only possible reason I can think is because some might not render it, but this seems unlikely, if it renders tex, it should render \empty. And \empty is listed on the tex mark-up page. {} is good for inline, but silly for general policy, it's nonstandard to say the least. Revolver 20:29, 8 Jul 2004 (UTC)

I agree. Dyspropsia was the one who made this change, citing "Wikipedia conventions." I realise some browsers don't support the character, which discourages using it in text, but the software should (although it may not be) be smart enough to send these browsers inline images instead. In display-mode math it should always be used, although I've seen the alternate notation in some texts. Derrick Coetzee 02:18, 9 Jul 2004 (UTC)

I like this article. I think it would be better of "(also not altogether unexpected, since AC itself is equivalent to asserting whether or not an infinite product is empty!)" were replaced with "(also not altogether unexpected, since AC itself is equivalent to asserting that infinite product is non-empty!)". But I've never edited anything on Wikipedia, so I'm suggesting this here instead of making the change myself. -David

I've added your change, but in the future, Wikipedia:Be bold - you can't hurt anything. Thanks for your contribution. Deco 17:30, 27 January 2006 (UTC)

The article says that Tychonoff's theorem is complex - what does that mean? (Certainly it is always difficult to rigorously use the axiom of choice. But is this theorem harder than other applications?) I do not think that it is necessary for Tychonoff to be complex - if the properties of ultrafilters (sorry, "maximal sets with the finite intersection property") used are established in advance then a proof of Tychnoff can be very simple. But I am not a topologist - I know very little. What do other editors think of the proof? 128.135.100.27 03:44, 11 February 2006 (UTC)

Well, complexity is subjective. Perhaps it would be more accurate to say that "Known proofs of Tychonoff's theorem from first principles of topology are complex", but even then you might not think it's really all that complex. Maybe this statement should just be removed. Deco 04:49, 11 February 2006 (UTC)

It might be good to add the fairly simple proof using ultrafilters. I might add it. I think the statement that the Tychonoff theorem should be removed, since it's fairly simple to state and simple to prove (although the proof relies on a lot of more basic results). 67.71.22.78 03:46, 16 October 2006 (UTC) Jordan

Should be totally reworked

In my opinion this article should be totally reworked. (I will do it myself when/if I get the chance.) Here are some of the issues I found:

1) The article contains an informal sketch of the proof of Tychonoff's theorem. Let me ask what is the purpose of an encyclopedia article on mathematics: does one come to wikipedia to find proofs of theorems? I think that substantial proofs (not just "explanations") should not be given in wikipedia: rather there should be references given to easily available primary sources, and at least one online source if possible. In this case, there is a nice writeup of Tychonoff's theorem on Henno Brandsma's "Topology Explained" webpage, and presumably in other places. Copying a proof out of Munkres seems unnecessary.

The point here is that energy spent on giving the proof would be much better spent on careful writing and also on inclusion of other information that a textbook is less likely to contain. In this article the historical section is too short:

2) In fact Tychonoff proved his theorem for powers of the unit interval [0,1]. The general proof (which follows the same ideas) was proved by Cech. Why was Tychonoff interested in this theorem and especially in the above special case? Presumably it is at least in part because one can use it to show that completely regular Hausdorff spaces (now called Tychonoff spaces) can be embedded in compact Hausdorf spaces, i.e., have compactifications.

3) The switching between Tychonoff and Tikhonov is annoying and distracting. After the first sentence, everything should be Tychonoff.

4) "For finite collections of compact spaces, this is not very surprising." If you think about it, asserting that something is not very surprising adds no information. If the reader was already surprised you are telling them something they already know; if not, you are delivering a sort of mild insult ("Oh, you were surprised? Well, you shouldn't have been.")

Much more useful here would be an explanation of why this case was in some form known much earlier. Namely, for metrizable spaces compactness is equivalent to sequential compactness (this is all well described elsewhere, and can be linked), and a simple diagonalization argument shows that a finite -- and even countable -- product of sequentially compact metrizable spaces remains sequentially compact metrizable. In particular one swiftly deduces Bolzano-Weierstrass / Heine-Borel in R^n from the version in R^1.

Notice that this also requires you to motivate why it is important to have the theorem for arbitrary products and not just countable products. (Also, the fact that arbitrary products of sequentially compact spaces are not compact should be mentioned somewhere.) One good reason is the Tychonoff embedding theorem, as above.

5) Saying that the theorem "depends heavily on the particular definition of the product topology" is vague -- many important theorems depend heavily on their hypotheses. It may be worth saying that the theorem does not hold for the box topology and therefore serves as one kind of justification of the correctness of the product topology.

6) In lieu of sketching the proof, a more worthy goal would be to try to enumerate the various "different" proofs. I know of the following:

(i) Tychonoff's original proof used the notion of "complete accumulation point", which is interesting insofar as this has all but been forgotten in the present day.

(ii) There is a proof using the Alexander subbasis theorem.

(iii) A proof using ultrafilters due to Cartan/Bourbaki.

[Note that Munkres' proof is a form of this one.]

(iv) A quite similar proof using universal nets, due to Kelley.

(v) A more "elementary" proof using nets, due to Chernoff.

"But in view of the complexity of the proof of Tikhonov's [sic!] theorem, and that mathematics can be completely modeled in set theory (i.e. the category of sets is a topos), this is not altogether unexpected." A lot of stuff is being summoned here to justify the completely unjustifiable point that Tychonoff's theorem implies AC is "not unexpected." This is POV and has no place in an encylopedia, and is also sort of disrespectful to Kelley -- I'm sure he was quite pleased when he proved the theorem.

The following two sentences can be deleted, because the sentence beginning "For example" makes the same point much more clearly.

Similarly, there is no need to give the proof here -- but you should certainly attribute the theorem to Kelley, with the appropriate reference (1950, Duke Math J. ...).

[Even if the proof were given, remarks about the opacity of Zorn's Lemma are POV, as is describing one part of the proof as "trickiest", and so forth -- this is not encylopedia-style writing.]

The fact that the pointless (i.e., locale-theoretic) version of Tychonoff does not require AC is quite interesting -- it should not be hidden in a commentary to a reference.

Finally, I disagree that Munkres' book is "a major general reference" and think that probably he too would disagree. Munkres is a very well-written introductory text. Standard references include Kelley, Willard, Engelking, and Bourbaki, among others.

-- P.L. Clark

First draft of rewrite

I rewrote the page according to the above comments. Detailed references and better formatting still need to be provided. Comments welcome. 72.145.124.24 23:02, 8 August 2007 (UTC)

Clarification of Edit Summary

I reverted PST's edit explaining that there are other important theorems in General Topology. I do not contest this; rather, I regard it as obvious. The point is that I provided a reference to one of the most well-regarded standard texts on GT which makes the assertion that Tychonoff's theorem is the most important theorem in GT. (This is not exactly what I said in my edit summary, hence this clarification.)

I happen to believe myself that Tychonoff's theorem is the most important -- it is used in many applications to algebra, analysis and number theory -- but that is not the point. In order to list any other theorem in the article on Tychonoff's theorem as being equally important, please give a sourced reference. Plclark (talk) 22:01, 17 May 2009 (UTC)

I do not contest your revert as I think you performed it with good intentions, and I myself am not sure whether my edit was appropriate. However, I do wish to make some points on the subject. To begin, point-set topology is a fundamental part of mathematics, as you know. It provides not only a basis for many other fields, but also is important in its own right (as a connoisseur of point-set topologist, I am allowed to say this!). However, it is also true that results such as "the conjecture that every normal Moore space is metrizable, is independent of the ZFC" will not have any significant impact outside of point-set topology. Despite this, I wish to stress that in my view, all results have equal importance irrespective of their applications. By "results", I mean non-trivial theorems in this context. If someone can prove a result such as the one referenced about Moore spaces, he must be as intelligent, if not more intelligent, that the person who proved Tychonoff's theorem. I sometimes find it unfair that people are not often credited for the work they do, because the theorems they prove have no significant impact outside their field. The point I wish to make is that I feel all fields of mathematics, and all results within these fields have equal importance. Whether this is universally accepted is a different matter; I still feel this way. I strongly agree that Tychonoff's theorem has the most number (if finite!) of applications outside of point-set topology and that your revert was completely appropriate. However, as I said, I do not completely agree with your reasoning. This is not to say that others will not either. --PST 11:28, 21 May 2009 (UTC)

Hausdorff property

There is one mathematical mistake in the whole article: compactness is Heine-Borel PLUS Hausdorff. The Heine-Borel property by itself is referred to as quasi-compactness. However, Tychonoff's theorem is that a product of quasicompact sets is quasicompact. This is issue is important for the proof that T. implies AC. Indeed, the topology given there on $B_{j}=A_{j}\cup \{x\}$ is non-Hausdorff (unless each $A_{j}$ is a singleton, in which case finding an element in the product set is a mere triviality!). Moreover, it would be nice to have a comparison of the proof using net with the proof using ultrafilters. (But maybe this should be done in the article about compactness.) —Preceding unsigned comment added by 192.16.204.67 (talk) 21:05, 21 April 2010 (UTC)

By convention, compactness does not imply the Hausdorff property on this wiki; see Wikipedia:Manual of Style (mathematics)#Terminology conventions. However, the same page also advises us that "each article should explain its own terminology as if there are no conventions, in order to minimize the chance of confusion," so perhaps the article should be amended. —Tobias Bergemann (talk) 13:29, 22 April 2010 (UTC)

What's the theorem, exactly?

The Definition section doesn't define anything. It accidentally implies that Tychonoff proved that the the product of sequentially compact spaces is compact (which isn't even true) with the statement that sequential compactness was more common in the 19th and 20th centuries together with the fact that his theorem was given in 1935 and the ambiguous opening sentence. In any case, there should be a clear, formal statement of the theorem and the current "definition" section should be "discussion" or something. 24.220.188.43 (talk) 19:30, 12 July 2011 (UTC)

Ultrafilter lemma and axiom of choice

Where do we use the axiom of choice and not only the (weaker) ultrafilter lemma in the proof? Only for creating a point in the product space converging in each factor? Does that mean in Hausdorff spaces we do not need the axiom of choice but only the ultrafilter lemma? (since limits are unique in Hausdorff spaces we would be able to use the replacement axiom instead of the axiom of choice to get a limit) --Chricho ∀ (talk) 22:14, 13 July 2012 (UTC)