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Added some material to end compactification (fixed minor errors, added an application). I had not been aware that the theory of ends was invented by Freudenthal in his dissertation.
Very nice, thanks! Does the “remarkable” fact in the hemicompact description essentially boil down to the fact that hemicompactness assumes a final subsequence of the directed poset of compact subspaces, so that limits over its opposite can equivalently be computed as sequential limits?
I didn’t write that part, but I think that’s exactly right.
(Unfortunately, there seem to be two different discussion threads about this page; the older one is here. I wonder if it’s possible to merge them?)
I’m confused by the “abstract” definition of the end compactification given. Let $X$ be a topological space, and let $Comp(X)$ be the poset of compact subspaces of $X$. The claim is that there is a functor $Comp(X)^{op} \to Top$ given by $K \mapsto K \amalg \pi_0(X - K)$.
First of all, what exactly does $Comp(X)$ mean? Does it consist of subspaces of $X$ which, in the subspace topology, are compact Hausdorff, or just quasicompact? In the latter case, are they required to be closed in $X$?
Secondly, how is this functor defined on objects, precisely? Does $\pi_0(X-K)$ mean connected components, or path components?
Thirdly, how is this functor defined on morphisms? For $K \subseteq K'$, the claim is that there is a map $K' \amalg \pi_0(X - K') \to K \amalg \pi_0(X-K)$. The restriction of this map to $K'$ is supposed to be the inclusion into $K$ for points that happen to lie in $K$, and map to the appropriate element of $\pi_0(X-K)$ otherwise. But this is not continuous!
Moreover, if $\pi_0$ means “connected components”, then we have to contend with the fact that $\pi_0$ is not a functor in general. I think this is okay, though, because we only need it to be functorial with respect to open embeddings, which I think it is.
Thanks for vetting this. It wouldn’t surprise me if there were some soft spots which could be fixed by adding “niceness hypotheses”. To answer one question: $\pi_0$ is supposed to be “connected components”. I think some authors might deal with locally connectedness (plus other hypotheses, perhaps), which I think would take care of functoriality.
I think some authors assume some separation condition like $T_2$ or $T_3$.
Of course this is all classical stuff in some sense, so everything can be repaired; the interesting question for me would be “in what generality can this be made to work?”.
The edit history says that I wrote most of this (about 6 years ago), but I don't remember doing so. (In fact, I looked stuff up on this page for a comment on Math Overflow without thinking that it was familiar.) But judging from the reference that I cited and my comments on the other thread, I got it from Wikipedia and never finished checking for myself that it was all correct. Now, Wikipedia has had that material for about 10 years now, which is a good sign, so I would assume Wikipedia definitions of compact (= quasicompact) and connected (not path-connected). But it's hard to be sure. The definition on Wikipedia was added by Jim Belk in this edit, so maybe we should ask Jim.
Thanks for the clarification!
Notice that the description by Jim Belk doesn’t actually describe the whole space – just the space of ends themselves. So the nlab article is the only place I’ve seen an attempt at defining the whole space, and this is where the most serious issue is, I think – for $K \subseteq K'$, the map $K' \amalg \pi_0(X-K') \to K \amalg \pi_0(X-K)$ simply isn’t continuous, unless $K$ is a union of connected components of $K'$.
It’s a very strange state of affairs – a priori, I would tend to assume it should be most natural to define the whole compactification in one go, and then simply take the space of ends to be the subspace of new points. But it seems that’s not the case!
I wonder if it could be modified with some trickery – like rather than taking the coproduct topology on $K \amalg \pi_0(X-K)$, take instead some coarser, possibly non-Hausdorff topology. Perhaps the quotient topology coming from the map $X \to K \amalg \pi_0(X-K)$?
If the end-compactification is compact Hausdorff and $X$ embeds in it, then $X$ is a Tychonoff space and admits a unique compatible totally-bounded uniformity whose completion is the end-compactification. So it seems to me that a natural way to define the end-compactification directly, without first defining the space of ends, would be to give a direct definition of that uniformity. In section 14 of The Shape of Infinity I described a gauge uniformity whose completion is the one-point compactification. I always assumed that there was a similar description of the end-compactification uniformity, using connected components of the complements of compact subspaces rather than their full complements, but I don’t think I ever checked it. And of course the definition could be reformulated using entourage or cover uniformities instead.
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