pi-base
closed interval with countably many origins
Hello, I've recently been browsing pi-base, and could not find a good example of a space that would be pseudocompact, not compact, $T_1$, and exhaustible by compact sets.
The space I am addressing in this issue is such example, see this post https://math.stackexchange.com/a/4935609/476484
I forgot how to do this, and additionally I'm going to be off for a week, but if no one is going to write a pull request then I will try to do it.
This is a variation on Line with countably many origins, modified so that the part away from the origins becomes compact, to get pseudocompactness. You could modify your example to be a circle with countably many origins, so that it also becomes Locally n-Euclidean (a non-Hausdorff one-dimensional manifold); this would allow to automatically deduce other properties, instead of having to deal with the endpoints of the interval $[-1,1]$.
Also somewhat similar to Telophase topology S65. I support adding it. I also support changing Telophase topology so that its definition makes it a subspace of line with two origins and the space you suggest. I'm interested in these spaces right now so I will make a PR for it soon.
@GeoffreySangston @prabau which example would it be better to add? The one I defined in my math.se post, the modification to a circle, or both?
@Moniker1998 I'll have to think about that. If the examples serve equal purposes then I would guess the locally Euclidean example, but I'll see if I can come up with any reason to prefer one or the other, or justify both.
Edit: Funnily enough, if the example prabau suggests was already in pi-base, then it's conceivable it could have helped me while working on the dissertation I'm writing. I thought a space I was studying was a 'circle with two origins' (actually remembering properly now, I thought it was a circle with two pairs of doubled origins), though it turned out to be a different non-hausdorff 1-manifold which is slightly more complicated. (The space I'm studying isn't a good fit for pi-base any time soon, but just barely.) So I'm personally biased in favor of adding the space @prabau suggests. I still want to think about the example @Moniker1998 suggests to see if I can try to justify to myself a reason to be interested in both.
Justification for a circle with two origins, by the way: π-Base, Search for compact + Locally $n$-Euclidean + ~$T_2$
For circle with countably many origins: π-Base, Search for pseudocompact + ~compact + Locally $n$-Euclidean + ~$T_2$
And technically a justification for @Moniker1998's example even if both of the above are added, though ~locally Euclidean, I would assume, is not a highly prized property by itself (not that this space is uninteresting.. and I still want to try to justify its interest compared to the circle form to myself, as I said above)
π-Base, Search for pseudocompact + ~compact + t1 + exhaustible by compact sets + ~locally euclidean
One kind of justification for adding @Moniker1998's example is then we can reference it in the definition of circle with countably many origins, which is analogous to how Long circle works.
Thank you, I'll wait for what @prabau has to say, and add both circle with two origins and circle with countably many origins, if no one has an issue with that (the justification for $[0, 1]$ with countably many origins is indeed pretty weak)
Yeah, I think the two variations of a circle should be fine. And I think we can skip the unit interval with countably many origins.
@GeoffreySangston Like you suggested, it would be good to mention that the Telophase topology is a subspace of the line with two origins.
@prabau @GeoffreySangston A circle technically doesn't have an origin, so would calling it a circle with countably many origins be fine?
Good point. What are synonyms in use for "line with two origins" and with many origins for comparison?
"Circle with a doubled point" for the two point case? Not sure about the case of countably many.
Unless the circle is moved so that the origin belongs to it. Or we can view one point of the circle as singled out as its "origin" ($e^0$), so maybe it's ok. I.e., circle = reals modulo integers. And we can treat the origin on the reals as its equivalence class in the circle, so it kind of makes some sense.
@GeoffreySangston @Moniker1998
@Moniker1998 I think it's an issue too. As @prabau suggests, 'Circle with a doubled point' works for that one, as in Everywhere doubled line. It's problematic because I'm also not sure what to call the countable variant.
I think I'd prefer not to use 'origin' for the circle, unless we find a reference (found two, neither of which cares to define it though) which does or cannot think of a better name which works for both spaces. It does have the benefit of signaling that the space is similar to the line with two / countably many origins. I don't think using 'origin' in the name is confusing per se, but more of a minor annoyance, and maybe it leads to kind of a useless chain of thoughts.
- Gelbukh's Topological dimension of a Reeb graph and a Reeb space has the term Circle with N origins, presumably for this concept.
- Also Gabard's paper has "circle-with two origins".
(The following is all fine, though I realized after assembling it that Circle with countably many branch points is problematic also.)
Haefliger and Reeb define the concept in general in Definition 3 on page 110 of their paper. They write point de branchement. Sorcar's translation of Reeb + Haefliger's paper (approved by Haefliger apparently) translates this as branch point in definition 3. The term 'branch point' seems to appear twice in David Gauld's book on non-metrisable manifolds, with this usage. It's not ideal since "branch point" has a more famous meaning.
Some paper's with branch point having this usage
- Mehidi's paper
- Forstnerič and Laurent-Thiébaut's paper. This paper has the term 'double branch point'. So maybe it's correct to refer to a double branch point and a countably infinite branch point? Double branch point also seems to be used in reference to branched coverings, however, and I don't think there a relationship here.
- Bhunia and Sorcar's paper (unsurprisingly)
- I think Thurston is using "branch" and "branch point" in this spirit in this paper
Maksymenko and Polulykah's paper notes that the Haefliger-Reeb paper calls this branch point, but chooses instead to call it special point.
"Branch point" also seems in use for a similar but different concept which is also related to foliations/laminations. See Shields's paper, where "branch point" is used to describe the points without a Euclidean neighborhood in a "branched surface".
So according to Haelfinger-Reeb definition, each origin of the line with two origins is a "point de branchement", right? It's not just for spaces like the branching line, which has a more real branching.
In any case, given the references you found with that name, I would be ok with "circle with ...-origins". One advantage is that it relates directly to the corresponding "line" spaces.
@GeoffreySangston What to you think of this: circle = reals modulo integers. We can treat the origin on the reals as its equivalence class in the circle, so it kind of makes some sense to call $1=e^0$ the "origin" of the circle in that manner. (and "origin" = identity element of the additive group of reals = identity element of the circle (as multiplicative group of complexes of modulus 1)
So according to Haelfinger-Reeb definition, each origin of the line with two origins is a "point de branchement", right? It's not just for spaces like the branching line, which has a more real branching.
Yes. Sorcar writes that a branch point $p \in X$ in a locally Euclidean space $X$ is a point for which there exists another point $q$ such that $p$ and $q$ are not separated by disjoint neighborhoods. Looking at Haefliger-Reeb, I think I confirmed that is what they define it as using my extremely limited understanding of French.
In any case, given the references you found with that name, I would be ok with "circle with ...-origins". One advantage is that it relates directly to the corresponding "line" spaces.
I support "circle with ... origins".
@GeoffreySangston What to you think of this: circle = reals modulo integers. We can treat the origin on the reals as its equivalence class in the circle, so it kind of makes some sense to call 1 = e 0 the "origin" of the circle in that manner. (and "origin" = identity element of the additive group of reals = identity element of the circle (as multiplicative group of complexes of modulus 1)
I deliberated with myself for too long now and I think it's okay. Will we refer to the 'origins' of the space in the trait files?
I'm kind of curious if there's a locally Euclidean space such that the Hausdorff reflection involves more than 1 step of gluing the non-separated (branch) points. Maybe one of the complicated examples from Haefliger-Reeb (Sorcar)?
The one-point compactification of the line is a circle. Similarly, the "circle with $\alpha$ origins" is the one-point compactification of the "line with $\alpha$ origins". (Something that could be mentioned in the description.) It's another good reason to go with that name.
Thank you, I'll wait for what @prabau has to say, and add both circle with two origins and circle with countably many origins, if no one has an issue with that (the justification for [ 0 , 1 ] with countably many origins is indeed pretty weak)
Do you still want to add these @Moniker1998? If you're working on other things, I don't mind going ahead and adding them with a few traits to get them started.