I was wondering if there is a theorem like
"If $f_i:X_i\to Y_i$ are homeomorphisms then $\prod_i f_i : \prod_i X_i \to \prod_i Y_i$ is a homeomorphism"
for $I$ finite. What about $I = \mathbb N$? Also, what for the same theorem with diffeomorphism instead of homeomorphism? I think I can prove it for $I$ finite thinking of product like a disjoint union. This would also work for $I$ infinite?