pi-base
Space (Alias) Suggestion: More aliases for Hilbert space $\ell^2$ (S30)
Space (Alias) Suggestion
S30 could have 'Unit sphere in $\ell^2$', and/or 'Closed unit ball in $\ell^2$', and/or closed half space in $\ell^2$ as aliases. Alternatively / in addition, the list of representatives in the description of S30 could include references to these.
Rationale
This is stated on this mo page.
Also, it just seems fascinating that these are all homeomorphic.
Other remarks
https://github.com/pi-base/data/pull/863#issue-2637246060 is a recent discussion about merging two spaces into S30.
https://github.com/pi-base/data/issues/829#issue-2615588856 is an 'Issue' thread associated to the previous thread.
Note that $S^\infty = \bigcup_{n = 0}^\infty S^n$ with the weak topology / topology coherent with $\{S^n : n \ge 0\}$ is not homeomorphic to S30 because it is not a Baire space, so I think $S^\infty$ wouldn't be a good alias.
This does sound interesting. I agree with adding these. I do wonder, out of curiosity, since $\ell^2$ is homeomorphic to any infinite-dimensional separable Banach space, does this result holds for those as well? Namely, is S30 also homeomorphic to the unit ball/unit sphere of all infinite-dimensional separable Banach space?
Hot take (so maybe I'm overlooking something): we might consider just noting these characterizations in the space description, rather than having a dozen different aliases.
In that case I'll write a PR adding these homeomorphic spaces to the description.
Ah, I've checked the reference in the MO post. As I suspected, the result about closed unit ball and unit sphere works for all separable Banach spaces. The only issue is the reference itself. I cannot find a DOI number of a zbMATH reference number for it, and I haven't been able to find any other reference for this fact. Is it possible to cite a reference with neither a DOI nor a zbMATH reference number?
There's an MR reference number for it: https://mathscinet.ams.org/mathscinet-getitem?mr=0478168
@StevenClontz
What's wrong with the following? Add/remove the most essential aliases (David added some more to the description.) It seems worth talking about rather than immediately shutting this down.
$\mathbb{R}^\omega$, $C([0, 1])$, Unit sphere of $\ell^2$, Closed unit ball of $\ell^2$
@GeoffreySangston I'm not sure if I understand the question, but if you're referring to my "hot take" above, the reason I called it a "hot take" is because I haven't considered it carefully. 🙃 So I don't think I'm trying to shut down anything.
@StevenClontz Okay, well thank you for that clarification. The question is, what should the list of aliases be for this homeomorphism class of spaces? As I understand it, the search only considers the aliases. The following only returns the circle right now:
https://topology.pi-base.org/spaces?text=sphere
(Aside: Maybe it should also return https://topology.pi-base.org/spaces/S000001, $S^0$, Zero-dimensional sphere)
It seems like there's a decently sized literature about unit spheres of Banach spaces. If it didn't have the misfortune of being homeomorphic to $\ell^2$, then the unit sphere of $\ell^2$ could be justifiably added as its own space. (It should not be, I think.) An alias would be the best thing, except there's the aesthetic limitation of wanting a short list of aliases.
A UI enhancement might be to only list the first three(?) aliases in tabular results.
Note that the description is considered, but isn't weighted heavily. From memory, I think we have name weighted 1, aliases weighted .7, and descriptions weighted .3.
I'm not well versed in this particular space so I'll rely on others in the community to decide specifics, but we should consider a general policy for how to handle a space that has several different names and characterizations.
General question. Is it really important to have all the possible non-obvious homeomorphic versions of a space listed as aliases?
Just to take separable Banach spaces, there are zillions of them. What's the harm if someone's favorite Banach space is not listed there? They will not be aware that their space is not already in pi-base. But most functional analysts will not care about that. And those who do care will probably know about Anderson-Kadec, so will have a good idea where to look. No harm done really.
On the other hand, the list in the description itself is pretty informative and nice to have.
@prabau
General question. Is it really important to have all the possible non-obvious homeomorphic versions of a space listed as aliases?
You're asking an extreme question. The answer is obviously not. No discussion required. The actual question which should be asked is, which of the aliases are worth having?
I apologize I was short tempered there. It was wrong of me to be so rude. I felt mischaracterized by @StevenClontz's first message which used the word "dozen", and your message which used the word "all". I think I'll log off for now.
Please, no need to apologize. My comment was extreme indeed, and you set me right. In any case, I think your summarizing question is a good one: which of the aliases are worth having?
Your suggestion of keeping
$\mathbb{R}^\omega$, $C([0, 1])$, Unit sphere of $\ell^2$, Closed unit ball of $\ell^2$
seems a good one. It is a list of a few cases, surprisingly non-obvious and interesting. And it's also short enough, thanks to the use of symbols.
I personally don’t have a strong preference either way, though I do find the list of aliases @GeoffreySangston is suggesting to be a reasonably good choice. Since this seems to be the consensus for now, I’ll add this to #975.