Jakob on Nostr: I just found a point set topology question in my notes which I asked myself years ...

I just found a point set topology question in my notes which I asked myself years ago. Maybe someone enjoys proving the claim or thinking of a counterexample:

Let X and Y be topological spaces. The product topology on X × Y certainly satisfies the following two properties:

1. The first projection p : X × Y → X is continuous and open.
2. For every x ∈ X, the subspace topology on {x} × Y agrees with the given topology of Y.

Question: Is the product topology uniquely determined by those two properties? If not, does the situation change if we assume X and Y are locally compact?

Regarding the relevance of the openness condition on p, see the comments by Dustin Clausen here: https://mathoverflow.net/questions/403487/is-there-a-notion-of-flatness-in-point-set-topology?noredirect=1&lq=1

Condition 2 means that the fibers of p are the correct thing and condition 1 says that the fibers vary continuously, so that hopefully we can spread out the correctness of the fibers to the whole thing.