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.