two theorems about weakly first countable

[felixpernegger]

1. Almost discrete + Weakly first countable => First countable

Easy, but we dont have this yet.

2. WFC + P-space => Alexandrov

This generalises T225.
Proof sketch: By P-space for every x there is an n such that any nbhd of x contains V_x(n). Then basically take the union of those to conclude alexandrov

[felixpernegger]

@prabau also, the implication WFC => Kolmgorov quotient is WFC really does not hold, I asked ChatGPT Pro and it came up with a fairly easy example.... (I checked it)
https://chatgpt.com/share/6a15e2b2-4944-832a-b93e-abc5015a2e1f

Would writing a warning about this for the metaproperty page be useful? (In my opinion yes, even though I dont think we need to add the space)

[prabau]

I completely agree it is useful to mention that a certain meta-property does not hold when the result is not obvious or the counterexample would be instructive. We have done this several times in the past:
https://topology.pi-base.org/properties/P000172
https://topology.pi-base.org/properties/P000140
and various others.

However, when we do this, we need to add the justification, mention an example or give a reference.
We should not add it without such a reference.
And a link to a private ChatGPT answer cannot be used.

How about this: Ask a question on mathse and see if some people give an answer. Please do not answer yourself. It's perfectly acceptable if nobody answers, there is no rush on this. Eventually somebody will.

You can also ask ChatGPT for a reference to that fact. Maybe it will find something.

In the mean time, we can keep this issue open.

[felixpernegger]

I agree with you (also I dont care too much either way) except one point:

How about this: Ask a question on mathse and see if some people give an answer. Please do not answer yourself. It's perfectly acceptable if nobody answers, there is no rush on this. Eventually somebody will.

It seems very odd to me to basically "withhold information" in a mathse post like this; we already know a counterexample, so asking other people to think about this problem to spend time to just solve it again is a bit strange.

---

I wanna note that there is a more general discussion in the math community on how to publish (correct) AI generated proofs right now; Submitting to journals is suboptimal (who is the author?) and even arXiV is quite restrictive.
As far as I know there is no consensus (or good solution) right now, so its probably not bad we ask this ourselves now for pi base.

@StevenClontz this might be of interest for you (meta question)

[prabau]

It seems very odd to me to basically "withhold information" in a mathse post like this; we already know a counterexample, so asking other people to think about this problem to spend time to just solve it again is a bit strange.

The goal is to encourage people to think about this problem and try to find an interesting answer. If we say we already asked ChatGPT and they gave us an answer, most people will just ignore the question. And if we give an answer ourselves, same thing: people will see it's already answered and will not bother.
Instead, we should feign ignorance, make the title of the post a question (Is ... ?) mention the known result one way and give its easy proof, mention that we tried to see how to prove the result that other way and were unable to. So is the converse true or is there a counterexample? All that will make people more interested in the post and we'll get better results.

(I routinely do this myself when posting questions on mathse. Very often I already know the answer and want to see what other people come up with. And the mathse community doesn't always like question+answer together.)

[prabau]

We want people to come up with better examples if possible, not just regurgitate what you already have.

[felixpernegger]

I understand your point (I dont agree with it; but thats not the point now), but, for the sake of a broader disucssion, imagine ChatGPT did in fact correctly prove that that there was an implication WFC => Quotient WFC and the proof is very tricky and hard to come up with.
How would you go about that? (And say if we made a post, no one answered it or whatever.)

It is worth asking ourselves this now, since its pretty likely to happen rather sooner than later given current trends.

[prabau]

Not a problem at all for the scenario you mention. If after multiple days/a week nobody has given a satisfactory answer, we can then go in and give an answer ourselves, mentioning that we asked ChatGPT and they suggested an answer along these lines ... and paraphrase it ourselves, as I think mathse has a guideline that we cannot just cut and paste AI output.

Note however that there is absolutely no rush to get a specific result proved and incorporated into pi-base "right now". These new results have been out of pi-base for years, so it's perfectly fine to wait a few more weeks. In the mean time, there are so many other things that can be worked on.

[prabau]

By the way, if you ask ChatGPT for a reference, does it come up with anything?

[felixpernegger]

[felixpernegger]

Its clear where the idea comes from (and once you have the idea its obvious it works), but nontrivial to think of it in the first place.

[prabau]

what is "quasi-first-countable" ?

[felixpernegger]

(I didnt read this)

[StevenClontz]

@StevenClontz this might be of interest for you (meta question)

disseminating AI proofs

I haven't found AI useful in proving things in general topology (I think most models are tuned to other areas of mathematics and maybe have implicit assumptions about topology that aren't true in general but hold in many common contexts). But if AI produced a correct proof, then I would still prefer that a human read it, write the proof in their own words, and then mention the assist. The act of re-writing the proof makes the contributor go through all the ideas carefully and hopefully decrease the chance of errors.

AI references

genAI is infamous for hallucinating references (random search result to this effect I did not read, random Reddit post related as well, new arxiv policy that has an extremely strong ban for hallucinated refs). This is something I wrote a recent grant proposal to solve in order to help with pi-Base model databases, but it was not accepted.

Of course, regardless of the method something is found, if the reference is sound, then there's no trouble. But one must treat this as a method of search, and treat any such results as potentially invalid/nonexistent

[felixpernegger]

I haven't found AI useful in proving things in general topology (I think most models are tuned to other areas of mathematics and maybe have implicit assumptions about topology that aren't true in general but hold in many common contexts)

In the last few months, it (expensive paid models at least) got significantly better. Its probably still stronger in combinatorics yes, but we shouldnt dismiss this.
Please see #1792.