Non T1 (P2) Toronto (P219) spaces

[felixpernegger]

I read the proof in
https://wrbrian.wordpress.com/wp-content/uploads/2012/01/thetorontoproblem.pdf
very carefully. There is a one typo (a missing +1), but the proof definitely is correct.

The Theorem in the paper is of course more powerful than just "Hereditarily connected", but we are missing the definition of having lower or upper topology (see the paper for definition) in pibase. They are also useful for not-yet-added theorems around Toronto spaces (Hausdorff + Toronto).
Do you think adding lower + upper topology would be worth it as a property (or some related concept?)? I'm a bit conflicted.

[prabau]

About lower and upper topology, it seems to be an ad hoc terminology for the paper. We have various examples of that in pi-base, where we call them left/right "closed ray" topology, or just "ray topology" when it happens to coincide with the "open ray topology" (generated by rays of the form $(\leftarrow,x)$ or $(x,\to)$). But I don't think we need to introduce this as a property in pi-base.

Note also that a space with a "lower topology" (i.e., admitting a total order such that the topology coincides with the corresponding "left closed ray topology") would be the same as "upper topology" by reversing the order.

[felixpernegger]

About lower and upper topology, it seems to be an ad hoc terminology for the paper. We have various examples of that in pi-base, where we call them left/right "closed ray" topology, or just "ray topology" when it happens to coincide with the "open ray topology" (generated by rays of the form ( ← , x ) or $(x,\to)$). But I don't think we need to introduce this as a property in pi-base.

Note also that a space with a "lower topology" (i.e., admitting a total order such that the topology coincides with the corresponding "left closed ray topology") would be the same as "upper topology" by reversing the order.

This is probably the right approach, yeah.
However note lower and upper topologies (in the paper) are only dealt with in the case of cardinals, so they are not really the homeomorphic (in that case).

[prabau]

However note lower and upper topologies (in the paper) are only dealt with in the case of cardinals

So it's even more specialized than I realized. Even more reason not to add this as a property.

[prabau]

On the other hand, if pi-base does not have it yet, adding a few examples of such spaces would not be a bad idea.

[Moniker1998]

Why do we exclude indiscrete spaces? They're clearly h. connected.

[felixpernegger]

Why do we exclude indiscrete spaces? They're clearly h. connected.

Because in the paper instead of "hereditarily connected", it is actually a very specific topology. But you are right, I overlooked this, will change

[Moniker1998]

@felixpernegger yeah. There is three of them and one of them is indiscrete. 🤔

[Moniker1998]

To be honest, I thought this property would be useless because of set-theoretical reasons.
But because we can consider it for non- $T_2$ spaces, it seems a lot better to me now.

[felixpernegger]

To be honest, I thought this property would be useless because of set-theoretical reasons. But because we can consider it for non- T 2 spaces, it seems a lot better to me now.

Well if you use explore, you can see it removes Toronto almost all non-T1 spaces.
As Under GCH all T2 infinite Toronto are discrete, the interesting examples are all T1, ~T2 (which actually exist)

[Moniker1998]

I haven't read this proof in detail, just most of the first part. The idea looks good. I'll let someone else review this.

[Moniker1998]

Do you plan on adding any results that would automatically deduce that $\omega_1+1$ is not a Toronto space?

[felixpernegger]

Do you plan on adding any results that would automatically deduce that $\omega_1+1$ is not a Toronto space?

I think so (since it is T2, you can check the Roadmap PR)? But I would prefer we first add all the theorems from the paper anyways before assigning traits manually, else it might get a bit messy

[prabau]

we first add all the theorems from the paper

I have not read the paper, but if the number of theorems is very large, it's perfectly fine to have multiple PRs with a few theorems in each. Please avoid monster PR that are hard to review. (Sorry for repeating myself again.)

[prabau]

For P196 (hereditarily connected), one other equivalent formulation for the definition would be:

• There is a basis of open sets that is totally ordered by inclusion.

That is easier to check in general, and is immediate for the spaces involved here.

Would that be worth adding?

[felixpernegger]

we first add all the theorems from the paper

I have not read the paper, but if the number of theorems is very large, it's perfectly fine to have multiple PRs with a few theorems in each. Please avoid monster PR that are hard to review. (Sorry for repeating myself again.)

I meant after this PR is merged I will make the next one, dont worry

[prabau]

Added the P196 characterization. How does that look?