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Did anyone ever write out on the $n$Lab the proof that for $X$ locally compact and Hausdorff, then $Map(X,Y)$ with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)
I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in
but of course there are more canonical references. I also added pointer to
It’s proved in [Munkres, Topology 2nd ed., §46].
Just to clarify, i didn’t ask for a pointer to a proof (I just gave two myself), but I just highlighted that the proof is not spelled out on the $n$Lab.
For some reason, nobody seems to be aware that Hausdorff is not needed here: it holds for an arbitrary locally compact topological space (defined appropriately). Hausdorff-ness only ensures that various definitions of local compactness coincide.
The definition of local compactness that I prefer (the proof also holds for it) is the following: for every $x \in X$ and every open neighbourhood $U$ of $x$, there is an open neighbourhood $V$ of $x$ such that the closure of $V$ in $X$ is compact, and is a subset of $U$. This is exactly the same pattern as the definition of local connectedness, with the exception that one requires that the closure of $V$ is compact rather than $V$ itself, which is the only sensible thing to do.
Thus I think that a proof would be useful to give explicitly. Perhaps I will find the time to add it myself at some point.
Nobody? :-)
Yes, “defined appropriately” is key (and we do have some material on this at locally compact space). What seems very relevant is that the topology should be a continuous lattice.
Hehe, it was not intended absolutely literally! I do find it rather amazing though that I don’t remember ever seeing ’in the wild’ (i.e. in something which is not directly concerned with characterising when a topological space is exponentiable) anybody write ’locally compact’ rather than ’locally compact Hausdorff’, even from people who one would expect to be careful about such details.
I taught a first course in topology for a couple of years, and, to me, it just seemed the obvious choice from a pedagogical point of view (actually for at least two reasons) to not impose Hausdorff-ness. Given the importance of exponentiability for locally compact topological spaces, I would expect anybody with a categorical or homotopy theoretic bent who has taught a similar course to have looked carefully at the proof, and to have made the same choice as me. But nobody seems to have done so, that I have seen!
In other words, my point here is really about the kind of point-set proof that one would teach to an undergraduate, rather than anything more sophisticated.
my point here is really about the kind of point-set proof that one would teach to an undergraduate
That’s exactly what I am thinking would be good to have displayed in the $n$Lab entry! If you have it written out in your preferred form, please consider adding it!
I do have some notes, which are actually not included in the notes that I have on my webpage, because I did not find time to polish them to the same standard as the rest. I will work on adding the proof to the entry now.
I have now added the proof to compact-open topology. I have not checked extremely carefully for typos, but hopefully the worst ones have been tidied up now.
I have not checked what is now Remark 15 (this was already present in the entry before my edits). In particular, I have not checked whether Hausdorff-ness is needed here.
Thanks, Richard!
I gave the prop-s and the corollary, a label, and then made the the pointers to them use that label to refer to it. This way the number gets hyperlinked and the number produced will remain correct no matter what other numbered prop-s are later added to the entry.
Thanks very much, my apologies for not doing this myself, I was a bit short of time.
I would also prefer the notation, remark, and definition environments to be in normal text rather than italics. Unfortunately, the only way around this would seem to be to use num_defn, etc, which will lead to an incomprehensible numbering system.
It would probably be easy to change instiki or whatever to use sequential numbering rather than numbering by environment. If this is desired, I could look into it when I get the chance.
I have added to compact-open topology (informally in the Idea-section) the remark that in the context of compactly generated topological spaces the definition is usually modified, etc.
Personally, I would love it if the numbering system were sequential throughout an entire page rather than individually sequential by environment type. I have my doubts that it would be so easy, or else we would have done it already, but if you want to work on that it would be great.
I do think that we should use the names of theorems that are provided, as is done on all other pages in the nLab; so I’ve taken the liberty of doing so at this page. Otherwise, if such a fix is ever implemented then we would have to go back manually and fix all the pages where we’d used the wrong names. And the incomprehensible numbering doesn’t really matter much when reading a web page where all references are hyperlinked anyway.
I’d want to keep as an option the numbering by environment.
Great, thanks for the reply. Then I will look into it when I get the chance.
I agree that we should keep the possibility of numbering by environment, and probably it would be best to keep existing pages as they are by default, and just manually change the others as people feel the urge. But we can discuss this if I figure out a way to change it.
Thank you for changing the environments, Mike. I did not know about num_note.
If you click on “Theorems” on the RHS of the edit page, it has a list and explanation of all the available environments.
Todd, do you actually prefer numbering by environment? That style always annoys me when I read it in a paper; I find that numbering all environments with the same counter makes it much easier to find things, especially things like definitions that are important and fairly rare. If the numbering is sequential, then when I’m referred to Definition 17 I know that it is between Lemma 16 and Theorem 18, but if it’s by environment then I have no way to guess that Definition 3 is to be found in between Lemma 6 and Theorem 2. Of course, with hyperlinked references this is less of an issue, but I still sometimes print things on paper to read. Is there an advantage to numbering by environment?
Mike, I understand your point. I’d rather not debate it, but for the nLab, yes, I personally like numbering by environment. In a paper or book, I find that numbering according to subsection is enough not to get lost.
I don’t want to have a debate either, but I do actually want to hear your reasons. I’ve never seen any advantage to numbering by environment, but I’m open to the possibility that there is one!
I fear the reasons might not make sense to anyone else because they are mainly personal and aesthetic. For one thing, the grammar seems off to me: Theorem 23 is not the 23rd theorem in the article. There’s also something about that style of numeration that begins to feel heavy to me, referring as it does to the mass of the entire book or article. (For a book or printed article, I much prefer a more modular system.)
Similarly, I don’t expect anyone to understand me if I say that I find German Fraktur letters heavy-looking and vaguely unclean. It’s mostly aesthetic: I don’t find “Corollary 45” attractive notationally.
Somehow the convenience you point out for a printed nLab article, which is where I’m most sympathetic to your argument, doesn’t trump these aesthetic considerations for me, and I’d prefer to keep what we have now as an option and even the default, at least for now. If proposed changes were to affect only the nLab proper, then I suppose I could get used to it in time, but I’d certainly feel very strongly opposed to losing control over my stylistic preferences in my personal web.
Just to remark that worrying about this point is moot as long as there is nobody in sight who might look into implementing anything either way. If we actually had somebody willing to do that, then we should think about the issue more broadly, also addressing cross-page reference identifiers etc.
I once was in position to have the time look into finding such a person. With Lee Worden and his “Working Wiki” we got to the point that Lee started to seriously present options, and it all looked extremely promising. (See the page he created here). Unfortunately, then life became more difficult for me and I don’t have the leisure anymore to look into this.
I feel quite sad and bad about this, because I left Lee alone right at the point where it would have become interesting. If anyone reading this here feels like he or she might be interested in and in better position to pursue this further, please do consider contacting Lee Worden again. Or maybe better, contact me and I’ll connect you to Lee (hoping that Lee might still be interested).
I am seriously interested in implementing it, Urs. I am currently working as a programmer. Just need to find the time, as in the little that I have I prefer to prioritise the mathematics itself; but I will definitely have a look within a few weeks at most, both at this and at referencing across pages. I am also interested in changes of a wider scope, but let’s start with something small.
Regarding Working Wiki, my inclination would be to instead start from scratch, and write something according to our needs. But we can discuss that when the time comes; I will just focus on some small tweaks to instiki first.
I have written an alternative statement and proof of the exponential property of the compact-open topology (here). Me personally, I find that proof easier to read than the one we have at compact-open topology. I was thinking of adding it as an alternative there, but I haven’t yet.
It is more or less exactly the same proof and statement as far as I can see. So I would use one or the other exposition at compact-open topology, not have both. I don’t mind the exposition which I gave at compact-open topology being replaced, it is purely a matter of style, and the style you have used fits better with the nLab as a whole, since you have written large parts of it!
Right, I didn’t mean to say that it’s another proof idea, just another style of writing it up.
Thanks, yes, looks correct now!
added pointer to:
added pointer to:
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