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I have written an article closed-projection characterization of compactness, so as to record a nice way of proving the Tychonoff theorem that is due to Clementino and Tholen. It’s rather direct and elementary, which doesn’t involve ultrafilters or nets or any such machinery. This might make it a possibility for a strong undergraduate classroom. (Munkres also has a proof which I haven’t cross-checked; the one I wrote up involves a smidge of categorical terminology, notably inverse limits.)
I didn’t want to stick in the proof at Tychonoff theorem as that article might be getting a bit bloated, but I did link there to the new article.
Thanks for this, Todd! That’s really very nice, in several ways.
Where you have
subbasis elements to consist of any subset of and
maybe it should read “…any open subsets of…”
and where you have
let us write by abuse of language
maybe one could just write ?
If you don’t mind, I’d be inclined to add double square brackets around a few more words, such as around “finite intersection property” but also around “Ioan James”.
Thanks, Urs! I’ll put in some more square brackets in a bit.
maybe it should read “…any open subsets of…”
I think it’s fine as is. We’re not trying to embed the space into for this proof. There are a number of different proofs of this type which in one way or another add “points at infinity” to the discretification of .
let us write by abuse of language
maybe one could just write ?
The reason I call it an abuse is related to the previous remark, that this union is at the set-theoretic, not topological level. Any suggestions for how I should put it? I can write instead of if that’s what you meant.
I’ve updated the article to include some variant proofs, which may help explain what is really going on. :-)
You made a really neat page there, Todd. I will have a closer look a little later, when my mind is back on the topic of compactness.
I took the liberty of embedding yet some more hyperlinks into the entry. But feel free to revert if you don’t like it.
Todd, I have now gone through your entry closed-projection characterization of compactness in detail, checking if I am following. In the course of this I added more hyperlinks and pointers to lemmas/facts used, in the hope that this will help the intended low-powered readership.
At some places I have ever so slightly expanded the proof text for clarity. For example at the end of the proof of the lemma for Tychonoff’s theorem I have highlighted that what we have shown is the contraposition of the claim. I hope that’s not inappropriately verbose.
Anyway, thanks again for this neat entry!
Thanks for your kind words! No, your slight expansions seem fine, and it’s nice to have another pair of eyes going over this. Let me know if anything seems unclear.
I still find it a little surprising how easily the inductive argument can be made to work – and particularly the argument at limit stages seems almost laughably simple.
On level of readership: generally when writing I try to shoot for a level that I think would be accessible to, oh, a second- or third-year graduate student who is interested in category theory. (Somewhere between being too hand-holding and too abstruse; that’s my personal taste.) But if you think this is understandable to a still wider audience, then good!
But if you think this is understandable to a still wider audience, then good!
Since, as you have seen me say, I have taken strategy to introduce only limits over “free diagrams”, I’ll use the proof for the special case of countable products (followed by an outlook remark for those interested that with just a tad more general kind of diagram, the general statement follows verbatim).
One could avoid the language of inverse limits altogether… although for a category theorist it’s a natural language, of course.
Added rather later: what I mean is that (referring to the proof of Tychonoff in the article, where one is considering the case where is a limit ordinal) instead of observing that is an inverse limit of the preceding , one can observe that in this case the collection of compatible lifts , ranging over all , do after all specify a unique element in , which we are calling . (The compatibility means that any two elements share the same coordinate in if .) The language of inverse limits is hardly necessary to make that observation – but to anyone used to limits it’s very natural to say it like that.
Pedagogically it’s not a bad idea to look first at the countable case , where the induction produces a sequence of compatible lifts
and thus a lift at the stage, which one shows to belong to the closed set . If the students get that, then the extrapolation to the general transfinite induction (going out beyond ) is not conceptually more difficult, and again you don’t really need inverse limits to discuss that.
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