My question is NOT “why does the product topology satisfy the conditions of a categorical product in Top”.
Rather, my question is, why does it do so uniquely? I don’t require necessarily a fully rigorous proof, but an conceptual understanding.
My guess is: If we take the cartesian product between two topological spaces $X,Y$, but we endow this product with a trivial topology $\{\emptyset, X\times Y\}$, then this also satisfies the categorical product conditions: For any topological space $Z$ and continuous functions $f:Z\to X, g:Z\to Y$, there is a unique function $h=f\times g:Z\to X\times Y$. We don’t need to put any more conditions on the topology of $X\times Y$ for $h$ to be unique. (This is essentially because the cartesian product $X\times Y$ is already “set-theoretically restricted” enough to ensure uniqueness of $h$). Therefore, we can put any topology on $X\times Y$, so long as it makes $h$ unique for all such $(Z,f,g)$. One topology that does this is the product topology, but another one that does it is the topology $\{\emptyset,X\times Y \}$.
What’s wrong with my argument?
EDIT: I Immediately realized that while this topology gets you a continuous $h$, it doesn’t get you continuous projections $\pi_X,\pi_Y$...