Toronto spaces - roadmap

[felixpernegger]

#1395 is meant to introduce Toronto spaces, however it is inactive (@StevenClontz).

Toronto spaces & the Toronto problem are relatively well documented in the literature, so I think it is reasonable to add them. All important known properties (as far as I can see) can be found in
https://wrbrian.wordpress.com/wp-content/uploads/2012/01/thetorontoproblem.pdf
also maybe see
https://en.wikipedia.org/wiki/Toronto_space

As stated in #1395 (and the link), under GCH we won't find T2 non-discrete Toronto spaces. Furthermore one can classify all non-T1 Toronto spaces (as some in the wordpress link). There might however still be interesting non-T2, T1 Toronto spaces.

I only added 3 trivial theorems so far. Adding the notion of upper and lower spaces would help eliminate many other spaces automatically, however I think this can be done at a later point. For now it would be nice to bring down the number of open, yet inactive PR's.

@Moniker1998

[prabau]

I like the branch name :-)

[Moniker1998]

This will possibly need a change in numeration of theorems, or some other PR will

[felixpernegger]

FYI I do think we should merge this eventually, but only after we find some theorems which automatically eliminate almost all spaces from being a Toronto space.

[prabau]

To keep things tidy, the related #1395 that started this has been closed.
We need to make sure any useful comments and references from that PR is being used here if useful.

[prabau]

I have not reviewed any of this (yet), but at a high level why do we have so many theorems here (14 and counting)?

[felixpernegger]

I have not reviewed any of this (yet), but at a high level why do we have so many theorems here (14 and counting)?

Because the paper
https://wrbrian.wordpress.com/wp-content/uploads/2012/01/thetorontoproblem.pdf
contains many of them, I just all took them. But I finished now

[felixpernegger]

There are some whihc can be deduced, I remove them now

[Moniker1998]

@felixpernegger did you read the proofs of those theorems you are adding?

[felixpernegger]

not yet, will do tomorrow, I need to sleep now.

[yhx-12243]

A good news: T915 and T917 are in fact redundant, though the reasoning chain is very long.

[yhx-12243]

Redundant, can be removed.

[yhx-12243]

Redundant, can be removed.

[yhx-12243]

Redundant after adding (Toronto + T₂ ⇒ Scattered) and #1562.

[prabau]

@felixpernegger My usual rant, after bad experiences in the past :-)
These monster PR loaded with non-obvious theorems are really not the best way to do things. Reviewing these can be painful. Much better to do it in multiple PRs, a few pieces at a time. Limited but steady progress gets to the end more quickly and safely.

And note it's irrelevant if for a new property most spaces have it as an unknown trait. Nothing forces us to "complete" spaces at the same time. That will happen eventually.

[yhx-12243]

One serious question is that, the original paper assume |𝑋| = ℵ₁. So we need to double check whether this result still holds without this condition.

At least I know some result (sequentially discrete, scattered) is holds for any T₂ Toronto space.

[felixpernegger]

@prabau if you prefer, I can split this up in 2-3 PR's

1. Definition plus trivial examples (finite, cofinite, discrete, indicrete)

2. Non T1 Toronto spaces

3. Toronto + Hausdorff

I think 1) is pretty much ready, 2) if we double check the proof of the paper and 3) still needs some work (see yhx's comment)

[prabau]

@felixpernegger I think that would be useful. We can keep this as Draft for now to refer to, and then eventually close it.

For comparison, see the comments we made for #1486. Not exactly the same situation, but somewhat similar. And unfortunately, that PR is not going anywhere soon due to these problems.

[felixpernegger]

Ok I will do that.

But about that PR I want to add, that I think this was handed very poorly. The author is very talented but due to the response he got on the PR likely won't ever contribute again.

[felixpernegger]

Completely unrelated but this is the longest deduction chain i have ever seen haha

[prabau]

But about that PR I want to add, that I think this was handed very poorly. The author is very talented but due to the response he got on the PR likely won't ever contribute again.

Yeah, it's unfortunate. We repeatedly told him not to do some things, but he kept going. Hope we'll be able to get back to his PR some day, once there is time to breathe with all the other PRs.

[Moniker1998]

I guess that S17 is Toronto will have to be added manually
Same with S19, S24 not being Toronto

[Moniker1998]

@prabau we should add new theorems based on this PR. Want me to do it? Or do we reopen this.

[prabau]

@Moniker1998 Yes, we should add theorems based in this "roadmap" PR. We can leave this PR closed, but have additional PRs for the remaining parts. If you want to do it, that would be great.

I have not looked at the details, but if there are more than a few difficult theorems, better do it in multiple PRs. That's the main reason we pushed back on the initial "monster PR" approach here.

[felixpernegger]

@Moniker1998 Yes, we should add theorems based in this "roadmap" PR. We can leave this PR closed, but have additional PRs for the remaining parts. If you want to do it, that would be great.

Just to make it clear, what remains of this PR which is not merged, is theorems about what a T2 Toronto space would look like (as GHC => Toronto + T2 + (some cardinality assumption) => Discrete), thus one wants to eliminate almost all T2 spaces for being Toronto, then maybe add rest manually.
As stated in an earlier comment, the paper always assumes $|X|=$ first uncountable cardinal (which I forgot to add in this PR), so one has to be careful which of the theorems generalizes to arbitrary infinite spaces.

One final thing which is not in this PR, as is not hard to show, S199, S200 and the space in #1591are Toronto spaces, and likely the only ones for which this is not super trivial (i.e. finite, indiscrete, cofinite or discrete), so the trait should definitely be manually added to those spaces (adding this also completes the explore search ~T1 + ?Toronto). :)

[felixpernegger]

So I have finally looked at the proof of Theorem 3.1 (on which the majority of the rest depends) in the paper. It seems correct, however unfortunately it actually uses the fact that $X$ has first uncountable cardinality. (namely it is used that if for some subset A is not countable, then A is homeomorphic to X for Toronto X, which cant be easily fixed)

One serious question is that, the original paper assume |𝑋| = ℵ₁. So we need to double check whether this result still holds without this condition.

At least I know some result (sequentially discrete, scattered) is holds for any T₂ Toronto space.

Proposition 3.11 (T2 Toronto Regular ~Finite => ~countably compact), Proposition 2.2 (T2 Toronto => Scattered), Part of Proposition 2.3 (T2 Toronto ~Countable => ~Lindelöf) hold anyhow. (+ maybe some others)

@yhx-12243 do you remember why sequentially discrete works anyways? (since this is basically Theorem 3.1)

[prabau]

@felixpernegger I am missing some things in your previous two comments above.
You say for example Proposition 2.2 (T2 Toronto => Scattered) holds independently of the cardinality of $X$. Why is that?

(Also, note that we cannot assume GCH in any of the pi-base theorems.)

[felixpernegger]

I would have to look at that again, but its just that the proofs easily generalisied to arbitrary cardinality.
EDIT: Though for scattered we may indeed need a regular cardinal or something?

[felixpernegger]

Closing this since we wont ever merge this.