Use of the term "compactification" on pi-Base

[StevenClontz]

Extending discussion coming out of #1159. I'm going to enumerate a few suggestions; please feel free to edit and add any alternatives I miss and correct mistakes I make.

1. "Compactification" is any compact space that contains the original space as a dense subspace.
   1. "A one-point compactification" is any compactification that adds a single point; there are several of these: https://en.wikipedia.org/wiki/Alexandroff_extension#Non-Hausdorff_one-point_compactifications So we must specify "the Alexandroff one-point compactification" for the Alexandroff extension (nbhds of new point are complements of closed-and-compact sets in the original).
   2. "The one-point compactification" always means Alexandroff extension.
2. "Compactification" is any Hausdorff compact space that contains the original space as a dense subspace. In this case, "the one-point compactification" is unique and only exists for non-compact locally compact Hausdorff spaces.

[StevenClontz]

I think I'm in camp (1.i). I've never assumed "Hausdorff" in my definition of "compact", nor "compactification"; I don't think I'm alone among working mathematicians. It seems more in the spirit of pi-Base to define things minimally, particularly with regards to separation axioms. Since there are many ways to "compactify" a space, being careful to always specify "Alexandroff [one-point] compactification" seems the most careful option.

[prabau]

That seems very reasonable and workable to me.

That matches the general definition of compactification in https://en.wikipedia.org/wiki/Compactification_(mathematics).
And as @GeoffreySangston mentioned in #1159 (comment), this is also backed up by some references in the literature.

[Moniker1998]

If we are going to make the change to 1.i, then we need to change all the names "the one-point compactification" to "Alexandroff one-point compactification" for all the Hausdorff spaces where the former is present. The reason being that uniqueness, even when the space is locally compact Hausdorff, is still not present.

As for the definition, this is a minor point, but I would specify that compactification is at its core really a dense embedding $i:X\hookrightarrow Y$ into a compact space $Y$. While $Y$ is referred to as compactification, this is really the pair $(Y, i)$ that is the compactification. Of course I realize that everyone probably knows that here, but it's still an important detail that can be easily forgotten.

So we are really not dealing with "spaces" as much as with "embeddings" and arguing if those embeddings should be into Hausdorff space or not necessarily, and what to name them in either case.

For me it's two different concepts, and I'd adopt either of them depending on my needs. I don't see a point in restricting myself to just one definition, but pi-base needs to be consistent.

It seems more in the spirit of pi-Base to define things minimally, particularly with regards to separation axioms.

It might be true for properties which one adds to pi-base, not necessarily for concepts that we use. And I think that's an important distinction. So this doesn't convince me.

Either way, I don't know of any ways in which non-Hausdorff compactifications are useful. So far I am not convinced that they are, no one has provided any solid reason to actually go for that definition, as far as I can see.

Could either of you @StevenClontz @prabau @GeoffreySangston provide reasons as for why those non-Hausdorff compactifications might be useful? And perhaps, why they might be useful for pi-base?

[prabau]

One general note. The notion of compactification is not any pi-base property, and does not appear in any pi-base theorems. It is just used in proofs and descriptions. But we don't need to give it a formal description in pi-base itself. It is just good to be somewhat consistent.

Question: If one starts with a Hausdorff space that is not locally compact, its "Alexandrov extension" is not Hausdorff. Would you also describe this case as a "one-point compactification" ?
(Engelking would not, because for him every compactification/compact space has to be Hausdorff.)

For your other comments:

Yes, it is true that we should then change many places to "Alexandroff one-point compactification". At least in the descriptions. Not necessarily in space names though, as it's often preferable to have a shorter name. The user can then read the description to see the full details, including this more expanded form.
This change is not urgent, but I agree it should be done, and there are probably not too many places anyway.
But maybe you have a point that it would be too much?

Good point about the embeddings. In many (most?) cases of interest, at least when describing spaces in in pi-base, we have a subspace $X$ of a space $Y$, so the embedding is just inclusion. But good to keep this in mind.

About the utility for pi-base, we should talk about the utility of the "Alexandroff extension", which is a synonym for "Alexandroff one-point compactification". It's just that "Alexandroff extension" seems maybe not so well known, whereas people are used to "one-point compactification" (For Hausdorff spaces, open sets are the complements of the closed compact sets in the original space. And the nice thing is that the construction for non-Hausdorff spaces is exactly the same. So no surprise for anyone.)

So why is that useful in pi-base?
We had introduced several spaces where we had a description, but could not find a short unambiguous name for it. Later, we realized it was exactly the one-point compactification of another space already in pi-base. That provided a short, clear, unambiguous name, ideal for pi-base.
https://topology.pi-base.org/spaces/all?filter=compactification gives a bunch of examples.
Now all these examples are compactifications of a Hausdorff space (though not all locally compact, so the compactification itself is not always Hausdorff).

But, here is the point, we can also add the case of the new "Circle with two origins", which is the "Alexandrov extension" of the "Line with two origins", which is itself not Hausdorff. And maybe other spaces later. So one-point compactifications of this sort can definitely be useful. And in some unknown future, if the deduction engine of pi-base becomes a lot more powerful, even that general case would become quite useful.

I agree that people who only care about Hausdorff spaces will not care about that.

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Possible alternative proposal (not necessarily convinced myself):

(1) If the space $X$ we start with is not Hausdorff, we should specify in full "Alexandroff one-point compactification".
Or even "Alexandroff extension (one-point compactification)" ?

(2) If the space $X$ we start with is Hausdorff, the expression "one-point compactification" will mean "Alexandroff one-point compactification".

But do we need to make a distinction between locally compact and not locally compact?

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After all my comments, I am willing to listen to other opinions and may be persuaded to change my mind.

Would like to get @david20000813 's opinion in particular.

[Moniker1998]

Question: If one starts with a Hausdorff space that is not locally compact, its "Alexandrov extension" is not Hausdorff. Would you also describe this case as a "one-point compactification" ?

I think, even though that there can be many (non-Hausdorff) one-point compactifications on a space, the Alexandrov construction is still the most famous one, so while I would change the descriptions in any case, I'd still keep the names of those spaces.

Either call "the one-point compactification" for Hausdorff compactifications (so description of S29 One Point Compactification of the Rationals would still need change) and Alexandrov one-point compactification/Alexandrov extension for non-Hausdorff compactifications, or be consistent and go with single "Alexandrov one-point compactification" in all cases.

Also note that not all "one-point compactifications" currently on pi-base are compactifications of a Hausdorff space, like One-point compactification of the metric fan

For other spaces, like Closed long ray they call it the unique one-point compactification. So even as of right now, pi-base is inconsistent when it comes to nomenclature relating to one-point compactifications

So yes, I don't think it matters if the original space is Hausdorff or not, but if the Alexandrov extension/compactification is Hausdorff or not, that is, if the original space is locally compact Hausdorff.

I kind of like keeping the name "the one-point compactification" or "unique one-point compactification" for (non-compact) locally Hausdorff spaces, and being a little inconsistent, referring to those cases when the original space is not locally compact Hausdorff as "Alexandroff/Alexandrov one-point compactification/extension"

That way, we let the context decide which particular definition of "compactification" we are currently using.

[StevenClontz]

Could either of you @StevenClontz @prabau @GeoffreySangston provide reasons as for why those non-Hausdorff compactifications might be useful? And perhaps, why they might be useful for pi-base?

For one, in https://math.stackexchange.com/a/5009157/86887 @david20000813 used the trivial one-point compactification (adding a focal point whose only neighborhood is the entire space) to produce an example of a [weakly locally] compact and semiregular space which is not Baire.

[GeoffreySangston]

Yes, it is true that we should then change many places to "Alexandroff one-point compactification". At least in the descriptions. ... But maybe you have a point that it would be too much?

There are only 15 files appearing under 'point compacti' when using the search feature on github_dev. Changing these to 'Alexandroff one-point compactification' would realistically take 5 minutes even without tools, so it doesn't seem worth worrying about. (It would also take 5 mins for the reviewer to double check no mistakes were made. But this is not worth worrying about.)

@GeoffreySangston provide reasons as for why those non-Hausdorff compactifications might be useful? And perhaps, why they might be useful for pi-base?

It's not clear to me if this is a question about terminology or about if these appear in legitimate mathematics. I really am agnostic on this, but a contrived answer to the first is that it appears in the name of the 'Wallman–Shanin Compactification', which applies to $T_1$ spaces and could appear in pi-base one day. As you say, context could be applied to allow either usage. I agree with what prabau has written in the section 'About the utility for pi-base'.

If this question is about the utility of these in mathematics, I won't make any great claims here but I'll just share some things I've learned recently. The non-Hausdorff cone construction (from the finite topological spaces literature, e.g. Barmak or Peter May) = open extension (from Steen-Seebach) = extension by a (closed) focal point results in a compact space. The non-Hausdorff cone is used in Barmak's book in a way similar to how a cone is used in conventional topology (there's also a non-Hausdorff suspension which is related to the non-Hausdorff cone in the same way the conventional suspension is related to the the conventional cone), and I believe that can be used to give a succinct construction of https://en.wikipedia.org/wiki/Pseudocircle and analogs of higher dimensional spheres). An expert in finite topology would have to be consulted to give you a real answer though. The open extension is known to be the minimal 'compactification' by one point, and the Alexandroff one-point compactification / extension is known to be the maximal.

My research is related to group actions. $\mathbb{Z}$ acts by homotheties on $\mathbb{R}^2$. The orbit space of $\mathbb{R}^2 \backslash \{0\}$ is called a Hopf torus, which is a basic example in flat affine geometry. The orbit space of of $\mathbb{R}^2$ is an extension of $T^2$ by a focal point. I don't know if this is useful or not, and I'm not saying it's amazingly important, but it helps me out at least by providing some kind of name and a non-zero amount of information (e.g., the orbit space is homotopically trivial). This kind of pathology seems to occur all of the time in geometry, e.g., the leaf space of the Reeb foliation of $S^3$ is the extension of $S^1 \sqcup S^1$ by a focal point.

[Moniker1998]

@GeoffreySangston "Wallman-Shanin compactification" that I found in Encyclopedia of General Topology seems to always be Hausdorff though, even though the construction is a bit more general as they mention.

They do mention that when the family is disjunctive, they still obtain a "$T_1$ compactification" but they only specify the Hausdorff ones by name.

Overall this seems to be exactly the same as the Wallman-Frink compactifications I was studying not too long ago

[prabau]

Yes, it is true that we should then change many places to "Alexandroff one-point compactification". At least in the descriptions. ... But maybe you have a point that it would be too much?

There are only 15 files appearing under 'point compacti' when using the search feature on github_dev. Changing these to 'Alexandroff one-point compactification' would realistically take 5 minutes even without tools, so it doesn't seem worth worrying about. (It would also take 5 mins for the reviewer to double check no mistakes were made. But this is not worth worrying about.)

I agree it's not much work to do it. When I was referring to "too much", I meant too much for readers to see, when the simpler "one-point compactification" would be adequate.

[prabau]

@Moniker1998
Forgetting about the one-point case for a moment.
Do you at least agree that the general term "compactification" can be applied the general case of a space embedded as a dense subspace of a compact space? (that is, without any Hausdorff assumption)

That is a well attested usage, and mentioned explicitly in Encyclopedia of General Topology for example, and other sources.

[Moniker1998]

Do you at least agree that the general term "compactification" can be applied the general case of a space embedded as a dense subspace of a compact space? (that is, without any Hausdorff assumption)

Yeah, of course.

It does make phrasing of certain theorems awkward, like "a space is Tychonoff if and only if it admits a (Hausdorff) compactification" but even if, that's not that much of a problem, and certainly not on pi-base.