Add Ellentuck topology S000223.

[JSMassmann]

I added the Ellentuck topology, a topology of cardinality continuum which is frequent in combinatorial set theory, and included references or proofs for its following properties: not normal; first countable; not ccc; zero dimensional; Baire; cardinality continuum; not locally countable; has no isolated points; has a coarser separable metrizable topology. Even omitting the "not ccc" and "not locally countable", no space in pi-base right now has this combination of properties, so I think it's valuable to add. All feedback is appreciated!

[prabau]

Thank you very much for this contribution. It seems a very interesting space worth adding to pi-base. (When adding a brand new space out of the blue, we often present it first in an "issue" to discuss if it's worth adding. But no problem here.)

General advice we give to all new contributors: We usually want our contributions to be not too large, breaking it up into smaller pieces if necessary. That makes the work easier to review and gradually merge into the main branch.

Anyway, will start looking at this shortly.

[prabau]

README: So, if I understand correctly, the condition \max(s) < \min(B\setminus s) is not absolutely necessary. Without it, one would get an equivalent base for the same topology. But specifying that extra condition provides a finer grained base, more convenient to work with.

In the Ellentuck paper you reference, he does not add this extra condition, right?
I assume this is what it calls the "Ramsey topology" in section 2 ?

Did Ellentuck introduce this topology in the referenced paper?

[JSMassmann]

I’ll keep all that in mind if I contribute again in the future.

I’m pretty sure that the condition is indeed not necessary but convenient. Now that I check the dates of publication, he seems to have first introduced it in “A new proof that analytic sets are Ramsey” which I cited for P64, but he did not give it a name there or study its intrinsic properties. In “Random isols”, he indeed calls it the Ramsey topology and does not impose that condition.

[JSMassmann]

I think one should be able to strengthen not having an isolated point to being homogeneous, but I don’t have a proof.

[prabau]

I assume you have experience with github? So for suggested changes like #1704 (comment) etc, if you agree with them, you should click on "commit suggestion". If you don't agree, you can reply in the comment thread to discuss further.

[JSMassmann]

Sorry, I'm familiar with most of Git but not reviewing, I didn't realize there was something I had to do myself regarding your edits. I've done so now.

[prabau]

No problem. There will be more suggestions :-)

[prabau]

In case you don't know, there is a "preview" window accessible from the Advanced tab in the pi-base web site (gets you to https://topology.pi-base.org/dev/preview), where you can paste raw content and see how it would display. Very useful when deciding how best to present something or for reviewing.

[prabau]

Apart from my suggestion above, $[s,A]$ becomes larger with a smaller $s$. So if $s$ is empty, $[s,\omega]$ would probably give the whole space. But if max(empty set) = 0, that does not quite work because all $B\in[s,A]$ would have to have min at least 1. Did you mean man(empty set) should be -1 maybe?

(I know it's irrelevant to specify a base for the topology. All that matters is what happens when the open sets become smaller. But just to be consistent.)

The problem is $\omega$ starts at 0. And on the other hand, the first sentence talks about "the set of infinite sets of natural numbers". That is in fact ambiguous, as some authors include $0$ in the natural numbers $\mathbb N$ and others don't.
So in the first sentence we could say:
"the set of infinite sets of nonnegative integers"
or maybe easier:
"the set of infinite subsets of $\omega=\{0,1,\dots\}$.

What do you think?

[JSMassmann]

Oh, that's a good point, I didn't consider it. I think we should write $\max(\emptyset) = -1$ instead. And I prefer the former of those two options for what to replace "natural numbers" with.

[JSMassmann]

Or one could avoid assigning a maximum to the empty set and replace $\max(s) < n$ with $\sup\{m+1: m \in s\} \leq n$, but that's not nice.

[prabau]

FYI, to display math braces in github comments, one has to use double backslashes: \\{...\\} instead of \{...\}.
(I just updated your comment to be able to read it)

[prabau]

I agree using max(empty set) = -1 is nicer.

[prabau]

For the reference, the paper "A new proof that analytic sets are Ramsey" (1974) https://zbmath.org/0292.02054 seems a better choice. It was the one that introduced the space, and that paper is more accessible too (available at https://www.jstor.org/stable/2272356).

What would be a good later paper, from any author, that also references that topology and that we could also add?

[JSMassmann]

While looking just now, I found that the Ellentuck topology is mentioned in Jech’s “Set Theory” (Def’n 26.25 to Cor. 26.38), which might be the best reference for this. It’s also talked about in Halbeisen’s “Combinatorial Set Theory” (Lemma 9.4 to Cor. 9.6), which redirects the reader to Matet’s “A short proof of Ellentuck’s theorem”, Carlson-Simpson’s “Topological Ramsey theory” and Mijares’ “Parametrizing the abstract Ellentuck theorem”, but I don’t know how useful those could be.

I originally learnt it from Wang’s “Complexity of codes for Ramsey positive sets” and Sabok’s “Complexity of Ramsey null sets”.

Also, one could move Plewik’s “On completely Ramsey sets” to the “main” bibliography? I’m not sure how this works.

[prabau]

These are great references. Adding Jech and Halbeisen is a good idea. So I suggest adding both to the refs: section and appending a paragraph at the end with the following:

Introduced by Ellentuck in {{zb:0292.02054}} (<https://www.jstor.org/stable/2272356>). See also Definition 26.25 in {{zb:1007.03002}} and the section "The Ellentuck Topology" on p. 248 of {{zb:1400.03002}}.

(That's Halbeisen 2nd edition for me, where it appears in ch. 10. Did you have a different edition?)

Can you add a commit with this and also incorporate what we discussed previously?

[JSMassmann]

I don't have a physical copy of Halbeisen, but the digital copy on his website, which I was looking at, has it in Ch. 9 on p. 214 (PDF 230). It's dated 2011, so presumably it's the first edition. Anyways, done.

[JSMassmann]

Sure, these are all reasonable.

[prabau]

P166 (has a coarser separable metrizable topology): I am not sure I understand the second half of it. But we can do it in a simpler way. It's kind of clear that the other topology is homeomorphic to the product topology on $\omega^\omega$ (with $\omega$ discrete), known as the Baire space. And pi-base already knows that is homeomorphic to the irrationals. See https://topology.pi-base.org/spaces/S000028 and the references there.

[prabau]

P50 (0-dim): I find the chain of either or or else ... a little confusing. Also the use of (\min(x \setminus (A \cup s)) + 1) instead of an interval like everywhere else is an unexpected dissonance maybe. Making a hopefully simpler suggestion below (check side by side in preview mode) and feel free to edit if needed.

[prabau]

P28 (first countable): the justification after the first sentence seems obvious. Do that really need to be there?

For the local base, would $\{[x \cap [0,n], x]: n < \omega\}$ be better? It's obviously equivalent if one interprets things in the right way. What do you think?

[prabau]

P29 (ccc): nice argument

[yhx-12243]

An offtopic question: Filtering all traits asserted currently, we found that S77 (Michael's product topology) satisfies all this nine traits.
Hence, are there some possibility that S223 is homeomorphic to S77?

[JSMassmann]

It's kind of clear that the other topology is homeomorphic to the product topology on $\omega^\omega$ (with $\omega$ discrete), known as the Baire space.

Yes, and the Baire space is in turn homeomorphic to $[\omega]^\omega$ as a subspace of Cantor space, through the map I exhibited there (sending $\{x, x+a, x+a+b, \cdots\}$ to $(x, a, b, \cdots)$), which is coarser than the Ellentuck topology.

I find the chain of either or or else ... a little confusing.

Yeah, sorry. I couldn't find an explicit proof that the $[s, A]$'s are clopen, so I tried to prove it myself, with results of questionable clarity (given my training as a set theorist, rather than a topologist). I'll commit your version (once I'm done writing this post).

the justification after the first sentence seems obvious. Do that really need to be there?

I thought it would be good to give some argument.

For the local base, would $\{[x \cap [0, n], x]: n < \omega\}$ be better?

Right, $x \cap n$ is under the von Neumann definition of natural numbers. So changing it to $x \cap [0, n)$, or $x \cap [0, n]$, seems reasonable.

Filtering all traits asserted currently, we found that S77 (Michael's product topology) satisfies all this nine traits.

P28, P29, P50, P64, P139, P166 are marked as unknown for S77. So, it doesn't satisfy all the traits asserted for the Ellentuck topology, it's just not known to not satisfy them... I'd be quite surprised if they were homeomorphic but you can't disregard it, I suppose.

[yhx-12243]

P28, P29, P50, P64, P139, P166 are marked as unknown for S77.

It can be proven. I'll make a PR about that when I'm on spare.

[JSMassmann]

Okay. Does S77 have the stronger form of Baireness which the Ellentuck topology satisfies, namely that the nwd sets form a $\sigma$-ideal?

[prabau]

Looking at Jech's treatment of this. On p. 524, equation (26.20) defines what is meant by $[s,A]$ (I am omitting the superscript) for $s$ finite and $A$ infinite. Informally, it's supposed to be the set of $X\in[\omega]^\omega$ s.t. $s$ is an initial segment of $X$ and the part of $X$ above $s$ is contained in $A$.

But (26.20) does not seem to be correctly defined? From $s\subseteq X$ one cannot conclude that $s$ is an initial segment of $X$. There could be additional elements of $X$ in $[0,\max(s)]=\max(s)+1$ and not in $s$.
What am I missing?

[JSMassmann]

That $s$ is an initial segment of $X$ is what the extra condition $\max(s) < \min(X \setminus s)$ - equivalently (together with $s \subseteq X$), $s = X \cap [0, \max(s)]$ - is for.

[prabau]

That s is an initial segment of X is what the extra condition max ( s ) < min ( X ∖ s ) - equivalently (together with s ⊆ X ), s = X ∩ [ 0 , max ( s ) ] - is for.

I know that. The condition $\max(s) &lt; \min(X \setminus s)$ is what makes things work, and what we are going to put in pi-base.

My observation was that Jech is missing that condition, or an equivalent condition. So I think Jech's definition of $[s,A]$ does not work. Would you agree?

[JSMassmann]

Oh, I see what you mean now. In 26.24, he makes the assumption $\max(s) &lt; \min(A)$, so that $X \setminus s \subseteq A$ implies $\max(s) &lt; \min(X \setminus s)$, and I can't tell where he tacitly uses that, but he does not explicitly write it again in 26.25, so yes, I would say it does not work.