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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Looking over the entry, I propose that Zoran’s comment is obsolete, but I also propose that what appears above it shouldn’t really be called a proof.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.
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