Improve definition Stone-Cech compactification of the integers (S108)

[felixpernegger]

No description provided.

[prabau]

I was actually also planning to update that description. So we may take the opportunity to modify a few other things.
Removed the easy label, as some discussion may be needed.

[prabau]

@felixpernegger @yhx-12243 @Moniker1998
Here are a few suggestions to be discussed. Would like to solicit your opinion.

• change the name of the space to Stone-Čech compactification $\beta\omega$ of the integers
  i.e., add the symbolic name of the space as part of the pi-base name

• Should we use $\beta\mathbb N$ or $\beta\omega$ for the main symbolic name?

• Have $\beta\mathbb N$ as alias and remove all the other aliases

[Moniker1998]

I'm not concerned. This is something @prabau will be good to decide on.

[felixpernegger]

@felixpernegger @yhx-12243 @Moniker1998
Here are a few suggestions to be discussed. Would like to solicit your opinion.

• change the name of the space to Stone-Čech compactification $\beta\omega$ of the integers
  i.e., add the symbolic name of the space as part of the pi-base name

• Should we use $\beta\mathbb N$ or $\beta\omega$ for the main symbolic name?

• Have $\beta\mathbb N$ as alias and remove all the other aliases

Probably using $\beta\omega$ is more consistent, but as far as I see $\beta\mathbb{N}$ is the only one that is being used in the literature, so I would be inclined to keep that and maybe even renane to Stone Cech compactification of the natural numbers?
But I suppose either way is fine. We should really have the Čech instead of Cech though :)

[Moniker1998]

but as far as I see $β \mathbb N$ is the only one that is being used in the literature

Maybe you haven't seen too much literature since $\beta\omega$ is certainly used just as often by specialists.

[felixpernegger]

ok my bad

[prabau]

@felixpernegger should I make a commit with my suggestions above?

[felixpernegger]

Yes your suggestions seem nice

[felixpernegger]

@prabau I applied your suggestions.

[prabau]

Why did you close this?

I'll take a look and possibly comment on some other things that could be added.

[felixpernegger]

Why did you close this?

I'll take a look and possibly comment on some other things that could be added.

As I said in #1591, I will take some time off from pi-base now, so I closed all issues and PRs opened by me.

[prabau]

One thing I don't like. We first say it's the Stone-Cech compactification .... And then the next sentence says: "Explicitly, $X$ is the set of ultrafilters ..."

The Stone-Cech compactification can be described in multiple equivalent ways. The description via ultrafilters works well for a discrete $X$, but it's not the only way. So we should mention something to that effect.

Maybe: "One way to describe $X=\beta\omega$ is as the set of ultrafilters ..." (no need for "all" in there).

@Moniker1998 do you think we should add anything?

[prabau]

As I said in #1591, I will take some time off from pi-base now, so I closed all issues and PRs opened by me.

Please don't close any of that. All the issues and PRs that you opened are valuable and can remain open. We can follow up on them and still work on them and bring them to completion.

[felixpernegger]

@prabau feel free to reopen any of them (just go to closed issue/PR list and order by recently closed). Especially the left ray $\omega_1$ is cool I think.

But I hope you understand that I don't really want to continue on this for now if someone complains about me allegedly dragging them down due to my "perceived feelings of persecution".

[prabau]

@felixpernegger I understand and I am sorry about all this. When discussions become too heated, it helps to have a thicker skin and take things more slowly. Your contributions and reviews have always been very valuable, so thank you for all of them. If some time in the future, after things settle down, you feel like getting back to it, you'll be more than welcome. And if not, I am sure you'll have plenty of things to keep you busy.

[Moniker1998]

But I hope you understand that I don't really want to continue on this for now if someone complains about me allegedly dragging them down due to my "perceived feelings of persecution".

That's not what I said. This was a message that if you have perceived feelings of persecution towards me then I don't want to deal with them. The if is important there, I was not making an assertion.

My position is completely reasonable.

And you have to understand that

1. Me being blunt is not part of my "aggression" towards anyone but it's my demeanor, and I apply it to everyone the same way. I act the same towards Patrick. It's not that I hate him or am angry at him.
2. I could technically stop being blunt and constantly consider how polite I should be etc., but for me this will just lead to reduction in productivity, because I did that before and it resulted exactly as I expected.

As Patrick said, you ought to have thick skin because this really isn't changing, and it's unreasonable to demand that it does. Making demands that I change my personality is really intrusive on my very own being. I understand you didn't see it as that though, but if you think about it, it really is that.

Lastly do note that I did accept your previous edits, I am not opposed to them. The other edit I had reasons to not accept.

If you look at how you acted as a result of that, you will also see that this is irrational behaviour. I understand that in your outlook you had a false positive of sorts, and you acted based on that false positive. This is the issue with emotions deciding for you, really. They're bad when it comes to false positives, and usually it's ill-advised to follow them.

That's how I see it.

[Moniker1998]

@Moniker1998 do you think we should add anything?

@prabau We could technically add the more common description of Stone-Cech compactification as the projective maximum of the compactifications of $\omega$, but it's not necessary.
The definition that is there now is my favourite one.

Alternatively we could define it as the closure of the image $\omega\to \mathbb{R}^{C^\ast(\omega)}$ where $n$ is sent to $(f(n))_{f\in C^\ast(\omega)}$, that's another construction. The projective maximum is more of a characterization in terms of properties.

The description looks fine as it is now to me though, this is just additional fluff.