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I gave locally compact topological space an Idea-section and added the other equivalent definition (here).
Would some non-examples be helpful? The rationals as a subspace of the reals seems to be most frequently mentioned. And infinite-dimensional Hilbert space.
A very typical example is, for instance, $\{ (0, 0) \} \cup \{ (x,y) \mid x \gt 0 \}$ with the subspace topology from $\mathbb{R}^{2}$. Proving that this example is not locally compact is an excellent exercise (uses several basic facts about compactness and closure).
Famously, locally compact spaces are not closed under infinite products in $Top$. They are closed under coproducts (of course), and under filtered colimits of open embeddings, as mentioned at ring of adeles.
Maybe #2 wasn’t so much asking for counter examples by themselves, as for their exposition being added to the entry.
I have added now the counter-example of countably infinite product of a non-compact space.
Yes, good to know both what and why some cases fail to satisfy a property. I added the rationals as a second counter-example. (About ’counter-example’, is it normally hyphenated? And isn’t it primarily used to refer to a case counting against a proposition, something we don’t have here? Or is there an implicit proposition its countering that all spaces are locally compact?)
Is there anything abstract general to say about local compactness? Cohesion meets compactness?
As for an abstract version of compactness, I see at compact object it’s noted that there’s a mismatch between compact object in $Top$ and compact space. Do we know the answer to Todd’s question there?
I don’t know if the story is any different for $X$ compact Hausdorff, but it could be worth considering.
About ’counter-example’, is it normally hyphenated? And isn’t it primarily used to refer to a case counting against a proposition, something we don’t have here?
How about “Nonexample”?
I don’t (normally) hyphenate “counterexample”. (My spell-checker didn’t balk when I just wrote that. Don’t know how a UK spell-checker would react.)
Looking over the entry, I propose that Zoran’s comment is obsolete, but I also propose that what appears above it shouldn’t really be called a proof. I’d be inclined to remove the proof environment, or else properly prove it instead of farming it out.
I’d have to think hard about my earlier comment about compact objects in $Top$ (from the Café), but I guess I’m a little skeptical that the situation would be any different for compact Hausdorff. For the moment I’m happy with applying Johnstone’s terminology to capture compact spaces (see compact element) and accepting that the compact object condition is probably too strong, although I also sense that the business about closed inclusions between $T_1$ spaces is a hack that could be improved somewhat.
I just found a webpage in which it states: ’OED does not include counterexample, but it does include counter-argument (note the hyphen).’ I have the shorter OED and it gives neither. The web OED gives the non-hyphenated form! Confusion.
FWIW: The ’counter’ is, I think, the English version of ‘contre’ in French, and in that language, the norm would seem to be ’contre-exemple’.
Yes, but perhaps ’nonexample’ is better, #8.
The OED has some catching up to do, obviously. But let’s face it: “counterexample” is generally a pretty math-y term. There must be thousands of recognized math terms that are not in the OED.
If to hyphenate or not to hyphenate is a US/UK thing, then so be it. Disputes over such things are so boring. Meanwhile, there is no disputing the fact that the unhyphenated (un-hyphenated?) form is very widespread.
I also think it is not important but here ’non-example’ (or nonexample) might fit the bill better. (I myself tend to hyphenate when a composite word looks too long for me or difficult to understand without the hyphen.) I was surprised that ’counterexample’ was not in the shorter OED as the word is quite often met in ordinary non-mathematical discourse.
The definition section at locally compact topological space used to essentially just point to Wikipedia for the issue of alternative definitions.
Now two alternative definitions are actually stated, and proof of their equivalence in the Hausdorff case is spelled out, here.
More alternatives (and their conditional equivalences) could be added.
I receive the following message regarding the entry:
I was looking at the change you recently made to the definition of a locally compact space in nLab, specifically definition 2.2. It’s a bit confusing the way it’s stated. It purports to be a suitable definition of local compactness for not necessarily Hausdorff spaces, but any space that satisfies 2.2 is necessarily regular (as is any space with a neighborhood basis of closed sets) and so nearly Hausdorff (except for the fact that it need not be $T_0$). An equivalent definition would be
2.2’ A topological space is locally compact if it is (pre)regular and every point has a compact neighborhood.
By this definition any locally compact $T_0$ space would be locally compact Hausdorff. It’s a perfectly fine definition as far as it goes, but it doesn’t really cover the non-Hausdorff case ($T_0$-ness aside).
Whoever it is seems to be correct.
But what’s the implication? I was following definition 2.5 at compact-open topology.
Off-hand I think that needs to be replaced with definition 2.1 of locally compact topological space. But I would need more time to look in detail at compact-open topology.
I think the point was exactly that 2.1 won’t work here in general.
And if I understand the complaint well, the suggestion is to amplify that 2.2, while not requiring exactly Hausdorffness, does require a whole lot. That’s fine with me.
Well, unless my memory is deceiving me terribly, it is a theorem that a locally compact topology in the sense of 2.1, being a core-compact topology, is an exponential object topology, which I though what compact-open topology would be driving at. I’m happy to take a closer look at the situation later. I suspect a subtle adjustment might do the trick.
My opinion is the same as Urs’. It is Definition 2.2 at locally compact topological space that is actually needed for proving that we have an exponential object, and Definition 2,2 is weaker than Definition 2.1. It makes no sense to me to assume Hausdorffness when it is irrelevant: Definition 2.2 to me is a very natural definition of local compactness in itself (also quite easy to motivate pedagogically, comparing with a typical definition of local connectedness, say), and tacking on Hausdorffness seems to me inelegant and unmotivated, all the more since it is entirely superfluous.
So, yes, as Urs writes, Definition 2.2 may still be implying a whole lot. But all those implications are completely irrelevant for proving that we have an exponentiable object and thus, since the latter is the main point of interest in locally compact topological spaces for a typical category theorist/homotopy theorist, i.e. the typical point of view of the nLab, my opinion is that Definition 2.2 is perfectly fine as it is, and that it is the correct definition with which to discuss exponentiability.
Richard, this is not a matter of opinion. It is a fact that a space is exponentiable iff it is core-compact, and locally compact spaces in the sense of definition 2.1 are core-compact. I refer you to Escardo-Heckmann; look at page 9 ff.
Now: whether the exponential topology is the compact-open topology – that’s a different story. As I say, a subtle adjustment to the compact-open topology may be required.
Even for Hausdorff spaces, neither of metrizability and local compactness imply each other, contrary to one of the examples claiming metric spaces are locally compact. The simplest counterexample is that of the irrationals with the subspace metric induced from the reals.
That should be taken out immediately. In addition to your example, a basic result from introductory functional analysis is that any locally compact TVS (over a local field like $\mathbb{R}$ or $\mathbb{C}$) is finite-dimensional, so that in particular any locally compact Banach space is finite-dimensional. (Fixed.)
One of the later examples (that of finite-dimensional manifolds) relied on the wrong example. I guess what was meant was something like “Euclidean spaces with their metric topology”, which would replace the now broken reference. I can come back to this later if no one else has done it in the meantime.
I took care of this too.
Thanks, Todd.
Hi Todd, I am not sure what you considered I was taking as a matter of opinion. I am perfectly aware of what you wrote in #22.
Yes, the topological spaces satisfying Definition 2.1 are exponentiable. But so are the topological spaces satisfying Definition 2.2, and Definition 2.1 implies Definition 2.2, whilst the converse holds only under Hausdorffness.
The person who wrote to Urs seemed to be claiming that there was something unsatisfactory about Definition 2.2. My point in #21 was simply that there is nothing unsatisfactory at all about it if one is interested in exponentiability, indeed, that it could be considered preferable for a typical mathematician reading the nLab.
It is of course standard to assume Hausdorffness as in Definition 2.1, but standard does not necessarily mean optimal from all points of view.
Hi Richard. To get one small thing out of the way, I started off by responding to what you wrote in #21: “My opinion is the same as Urs’.”
Definition 2.1 implies Definition 2.2, whilst the converse holds only under Hausdorffness.
Don’t you mean the other way around? Definition 2.2 is stronger than Definition 2.1. Definition 2.1 does not assume Hausdorffness.
The person who wrote to Urs seemed to be claiming that there was something unsatisfactory about Definition 2.2.
I think the complaint was that definition 2.2 was stronger than definition 2.1 and that it comes very close to asserting Hausdorffness.
Anyway, I’m glad you now agree that definition 2.1 guarantees exponentiability, although this seems to contradict what you said in #21:
It is Definition 2.2 at locally compact topological space that is actually needed for proving that we have an exponential object
which was the precise point where I felt the need to reassert myself.
But my larger point was in #20: that the article compact-open topology is fine and dandy, but seeing that it uses the stronger definition 2.2 even though the weaker definition 2.1 suffices to guarantee exponentiability, my suggestion is to augment compact-open topology by explaining the adjustment one makes to define the exponential topology under 2.1. Since the entire point of the compact-open topology is to discuss exponentiability, this augmentation to the article would seem to be very reasonable.
Or, an entirely new article could be written, giving a proof of this (stronger) result that locally compact spaces under 2.1 are exponentiable, and then that article should be linked to from compact-open topology.
Core-compactness is already discussed at exponential law for spaces. Probably that should be hyperlinked better with compact-open topology and locally compact topological space, and maybe even renamed or merged with something.
Hi Todd, my apologies, I had not read closely enough, and what I was thinking was Definition 2.1 was in fact some other definition which implies Definition 2.2, with the converse holding only under Hausdorffness. In other words, you can disregard most of what I wrote, as it was not referring to what is actually Definition 2.1… and I now understand and agree with your point!
I do still maintain my point that any separation-like axioms that Definition 2.2 implies are irrelevant from the point of view of exponentiability, and that Definition 2.2 is a perfectly fine definition, but I completely agree with you that it would be great to give a direct proof that Definition 2.1 implies exponentiability as well.
My main concern was to avoid making Hausdorffness (or $T_{0}$-ness, or whatever) part of the definition of a locally compact topological space, and this you were not disputing I now see :-).
Okay, I see that the proof of the continuity of the evaluation map only invokes def. 2.1, not def. 2.2, and never did otherwise (right here). Thanks for the input on that.
But I am still not sure what the forwarded message in #15 is suggesting should be changed. This message is referring just to the entry locally compact topological space, not to that on the compact-open topology. I re-read the Introduction and Definition section, but I don’t see which wording regarding Hausdorffness or non-Hausdorffness is regarded as confusing. I guess I’ll go an check with my correspondent.
I think that the forwarded message #15 is objecting to the emphasis placed on the fact that 2.2 does not require Hausdorffness, i.e. is suggesting a more precise discussion of what is implied by 2.2 when it comes to separation axioms.
That’s what I originally thought, too. But re-reading the entry, I don’t see which words should be changed. (I have tried to check with my correspondent, but no reply yet).
On the contrary, to me what the entry is lacking is statement of yet further variants of the definition which are in use, and which all agree in the Hausdorff case.
BTW Richard, are you going to weaken the assumoptions at compact-open topoogy? It’s your proof there, I would hesitate to mess with it.
To help sort this out, I have written up an article on my private web that gives a succinct proof of the most general fact here, that core-compact spaces are exponentiable.
So here would be a possible plan for compact-open topology. First, scrap the definition of local compactness (definition 2.5) that is currently there, and replace it by the weaker definition 2.1 from locally compact topological space. Then observe that locally compact spaces in that sense are core-compact, and then copy over the proof I wrote up. Also, maybe change the name of the article to exponential topology and make compact-open topology redirect to it. Finally, include some material that indicates when the exponential topology is the compact-open topology – this may be where we resurrect the stronger definition 2.5 for a more specialized purpose.
Then, over at locally compact space, either scrap definition 2.2 (since it is not equivalent to 2.1), or else move it to a section where various formulations in the Hausdorff space are considered.
A good example of a locally compact space which doesn’t satisfy the definition 2.2 formulation is any infinite set with the cofinite topology (that’s $T_1$, but far from Hausdorff).
Regarding #35: I have just made an edit, Urs, to locally compact topological space, which I feel addresses the point raised by your correspondent. Of course, as you suggest, one could add other definitions, such as the one the correspondent proposed.
Regarding the proof at compact-open topology: I do not intend to change it, as I think it is valuable to have a proof proceeding directly from 2.2. This is a very classical and elementary argument, and it can be useful to have such proofs, I feel.
However, this does not of course prevent others from making whatever changes they like. For instance, one could give a proof proceeding directly from 2.1 in addition to the one already at compact-open topology. It could for instance be that, since it is stronger, the proof from 2.2 is somehow more straightforward than that from 2.1, or illustrates some useful point which is not as clear as a proof from 2.1. I would be entirely happy with a proof from 2.1 (or from core-compactness, or whatever) coming first, with the one from 2.2 left at the end. But I would at least keep the latter.
Thanks Todd. For what it’s worth, I think the proof that is presently at compact-open topology would be adapted simply by replacing 2.5 by the weaker definition, since it is only the weaker form that is actually used further down in the proof.
But as far as I am concerned, please feel free to edit either entry.
Perhaps just to clarify a little more, my main goal with Definition 2.2 was to find the simplest way to prove exponentiability which was suitable for teaching in a first course in topology, and which did not assume Hausdorffness. All topological spaces of interest to a geometrically flavoured course are likely to be Hausdorff, so the point is not the generality, it is to emphasise that the Hausdorffness is completely irrelevant when it comes to exponentiability.
But if what Urs writes in #38 is the case (I haven’t checked), i.e. the proof at compact-open topology is exactly the same using 2.1 instead of 2.2, then what Urs proposes in the first paragraph #38 should be perfectly fine.
If core-compactness is brought in, I’d do that in addition to the proof at compact-open topology (possibly with 2.1 instead of 2.2 if that goes through), rather than instead of it.
Urs, I’d like to go through your proof carefully. Off-hand I’m a little bit skeptical if the assertion is that the exponential topology matches the compact-open topology in quite that generality.
Okay, so the immediate question that comes up is how you get step 2 in the proof of 3.1. It doesn’t look true for an infinite set with the cofinite topology, which is locally compact in the sense of definition 2.1.
But do we need later on that the compact neighbourhood in that step 2 is a closure?
I believe that I agree with Urs, that the proof of 3.1 at compact-open topology goes through perfectly well using 2.1 at locally compact topological space instead, and this is the only place that local compactness is used. So I am happy to use 2.1 instead of 2.2 for that (but it would be nice to make a remark saying that one can use 2.2).
PS - I corrected a typo and fixed a reference at compact-open topology.
Actually, now I agree as well. All you need to do is replace the statement by an interpolation of type $U \subseteq K \subseteq V$, and I think it’s fine. So very good.
Thanks to you two for checking!!
It seems you are leaving the task of editing the proof in the entry to me. I did so now: removed the extra clause about forming the closure of $V$ in def. 2..5 here, so that now $V$ is already a compact sub-neighbourhood. Then I replaced all following occurences of $\overline{V}$ with $V$.
Notice that the writeup and style of the proof is very much Richard’s. I tend to feel I get a bit lost in the notation there. I feel more comfortable with my own wording of Richard’s proof, here. That’s just to say: better check my edits (as long as nobody else edits the entry, these edits are highlighted in green and red here).
Hi Urs, unless I’m missing something, it’s not quite right yet. I’ll try to fix it now.
I have fixed the proof of 3.1 at compact-open topology now. Note that when we say that every neighbourhood has a compact neighbourhood in Definition 2.1 at locally compact topological space, it is vital in the second use of neighbourhood that we mean that it contains an open neighbourhood of $x$, not just that it is any old set which contains $x$. Otherwise the proof of 3.1 does not go through (one cannot take $U''$ to be $V$, which was what was written before I edited just now).
By the way, Urs, I think the same issue affects your proof at Introduction To Topology, you may wish to fix it there.
As I think I mentioned once before, I am entirely content with replacing the proof I wrote at compact-open topology with your re-write :-).
Not sure what you are pointing me to. I say “there is a compact subset which is still a neighbourhood” and “neighbourhood” implies “contains an open neighbourhood”. But I may be missing what you are trying to tell me.
In the proof of continuity of the exponentiation map at Introduction to Topology – 1 that you linked to, it is claimed that $K \times V^{K}$ is open. This does not parse: you would need $K$ to be both compact and open, and local compactness does not give that.
I.e. you need to replace $K$ on the left of the product by an open neighbourhood of $x$ which it contains.
Thanks!! I have fixed it now.
You’re welcome! Nice one!
Isn’t the Heine-Borel property just that closed and bounded implies compact? This doesn’t imply local compactness.
In any case, I am quite sure that locally compact Hausdorff TVS are finite-dimensional. For a source of this claim outside the nLab, see for instance here.
You may be thinking that there are (convex) bounded open neighborhoods of the origin in your Frechet space example, whose closure would also bounded. But that’s just not so: a TVS with such neighborhoods is normable, which is not the case with your example. See theorems 7 and 8 here.
Thanks, Jürgen. Yes, I agree this entry is probably due for a clean-up.
added the statement (here) that: Every locally compact Hausdorff space is compactly generated and weakly Hausdorff.
That would imply that a Hausdorff k-space is a colimit of compact Hausdorff spaces, no? Locally compact Hausdorff spaces are colimits of compact Hausdorff spaces, as noted on locally compact topological space just above where you inserted the statement there.
You can drop the Hausdorff assumption: Escardó, Lawson, and Simpson (doi:10.1016/j.topol.2004.02.011) state that a space is a k-space iff it is a colimit in Top of compact Hausdorff spaces (lemma 3.2.iv, p. 110 = p. 7) iff it is a quotient of a locally compact Hausdorff space (definition 3.3 and corollary 3.4.iii, p. 111 = p. 8).
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