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As a kind of supplement to Urs’s running topology series, I wrote an article colimits of normal spaces. Mainly I had wanted to write down a reasonably clean proof of the fact that CW-complexes are $T_4$ spaces, in particular Hausdorff, as called for on the page CW-complexes are paracompact Hausdorff spaces, but working in slightly greater generality. There are a whole bunch of links to stick in, which I plan to get to.
This page has taken me longer than I had first anticipated. Only after some struggle and reading around did I discover the power of the Tietze characterization of normality, which can be used to give a simple proof of the following general fact:
If $X, Y, Z$ are normal and if $h: X \to Z$ is a closed embedding and $f: X \to Y$ a continuous map, the attachment space = pushout $W = Y \cup_X Z$ is also normal.
This doesn’t seem so easy to prove with one’s bare hands (i.e., just using the usual definition of normality and reasoning away)!
Urs, after recent discussion with Richard about paracompactness, where do matters stand on the page CW-complexes are paracompact Hausdorff spaces? It would be nice to tie up whatever loose ends are still left hanging there.
Thanks, Todd!
where do matters stand
[edited:] So I have replaced the brute force proof for single cell attachment and added in the immediate corollary that countable CW-complexes are paracompact: here
If you could improve on that situation that would be nice. Please feel free to edit the page!
By the way, is there any problem with the article Miyazaki 52?
I haven’t really looked at it, but somehow it’s not being cited. In the MO-discussion here one person first pointed it out and then deleted that comment.
(Maybe that’s just because the OP seems not to be asking about paracompactness of CW-complexes, which he seems to take for granted, but of that of spaces of the homotopy type of a CW-complex. On the other hand, he didn’t complain about Neil Strickland’s pointer to Fritsch-Piccinini 90, which was posted on the same day as the (deleted) pointer to Miyazaki 52.)
Yeah, that’s the kind of quick and clean and elegant argument that I really like (what you put in for countable CW-complexes)!
I haven’t looked at Miyazaki either, but I will say that Fritsch-Piccinini is a fantastic reference. (That’s where I picked up on using the Tietze extension characterization that I mentioned.)
So just to let you know: I looked yesterday at some of the approaches to proving paracompactness of all CW-complexes. The approach via E. Michael’s selection theorem(s) seems attractive, and I think it has nPOV potential as well, but I want to understand it better. On the other hand, it may be worth contemplating the choice principles involved; the use of Zorn’s lemma by Michael may be overkill. Still weighing options.
Thanks, Todd. You are closing a real gap in the literature here. Not regarding the result, of course, but regarding a decent proof of it.
Well, I am gradually becoming convinced that a completely parallel story can be told for paracompact spaces as was told for normal spaces on the colimits page, and all of this fits neatly within the general story that Misha (Gavrilovich) has been telling us, that many concepts in topology can be explicated in terms of lifting/extension properties. What the Michael selection theorem gives is a characterization of paracompact spaces in terms of an extension property (it is based ultimately on the characterization in terms of existence of partitions of unity), just as there is for normal spaces, and I think this type of characterization makes all those colimit results fairly easy to prove, in completely analogous fashion.
Michael’s paper is from the mid-50’s and doesn’t use explicitly categorical language, but that language is really there and deserves to be brought out.
Unfortunately I am busy with other things for most of the rest of this day…
Sounds great! There is no rush.
Okay, I’ve finally written up something at colimits of paracompact spaces, mirroring colimits of normal spaces. I was delayed somewhat by being stuck for a while on an “it-is-easy-to-see” statement in Michael’s continuous selections paper, but anyway a pretty succinct proof of paracompactness for general CW-complexes is now there for your perusal.
And to start bringing matters to more of a conclusion, I performed some more edits at CW-complexes are paracompact Hausdorff spaces.
At some point I may write up some details of Michael’s selection criterion over at Michael’s theorems. Urs, I think you may have looked into this yourself already, right? You were looking for example at some of Akhil Mathew’s exposition(s). E. Michael as you know also proved that closed quotients of paracompact spaces are paracompact, which sounds sort of innocent at first blush, but it’s really not. I have not delved into this yet.
But a $T_1$ paracompact space admits partitions of unity if and only if it is Hausdorff… (Wikipedia)
Okay, thanks; I’ll look over this (and look over again at Michael’s paper, where I got the argument from). It’s not a seriously big deal for the current application, since it was already established that CW-complexes are $T_4$.
David, I can’t tell which precise part of the Wikipedia page you are looking at. Meanwhile, I checked Michael again in two papers, and he says $T_1$ in both of them.
Actually checking up on all this would involve some serious chasing down rabbit holes, but it seems to me that a space with a cofinite topology might be a kind of acid test. (That’s compact and therefore paracompact, and also $T_1$, right? But not Hausdorff.)
Well, from what I can tell according to this paper, theorem 4.6, page 7/12, “paracompact” for Michael must mean both paracompact and Hausdorff (perhaps in the same way that for some people, “compact” means “compact and Hausdorff”).
Assuming this is the case: even better! In that case, Michael proves that if $X$ is $T_1$, then paracompactness+Hausdorff is equivalent to his selection criterion. Now, one can prove by very soft means that CW-complexes are $T_1$, meaning that the whole business with normality and Hausdorff can be dispensed with altogether, and Michael’s selection criterion then allows one to prove CW-complexes are paracompact and Hausdorff in very short order.
The relevant sentence is
a $T_1$ space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover
But it doesn’t give a proof or reference.
But it doesn’t give a proof or reference.
That’s okay. I think it must be right, because now I’ve found confirmation in Michael. In A Note on Paracompact Spaces, Michael defines “paracompact” on the first page to mean Hausdorff and every open cover has a locally finite open refinement, so that’s the definition he works with when he proves his theorem (which starts by recalling the equivalence with the subordinate partitions of unity property).
Glad it got sorted out :-)
Thank again, Todd! Very nice indeed.
Re #20: that is quite amazing, thanks! At a quick glance, $Z$ seems to be the pseudocircle? I don’t quite know what to make of that, but I find it fascinating!
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