Talk:Tychonoff's theorem

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title: Untitled

message: 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)
thread_title:  First draft of rewrite 

message: 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)
replies: text: 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)
thread_title:  Clarification of Edit Summary 

message: 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)
replies: text: 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)
thread_title:  Hausdorff property 

message: 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)
thread_title:  What's the theorem, exactly? 

message: 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)
thread_title:  Ultrafilter lemma and axiom of choice