I can't say that it's better than o1 for my needs. I gave R1 this prompt:
"Prove or disprove: there exists a closed, countable, non-trivial partition of a connected Hausdorff space."
And it made a pretty amateurish mistake:
"Thus, the real line R with the partition {[n,n+1]∣n∈Z} serves as a valid example of a connected Hausdorff space with a closed, countable, non-trivial partition."
o1 gets this prompt right the few times I tested it (disproving it using something like Sierpinski).