Does Product Topology Distribute Over Disjoint Union?

Question

Is it generally true that if $X$ and $Y_i$ are topological spaces, then we have the homeomorphism: $$X\times\left(\bigsqcup_i Y_i\right)\cong\bigsqcup_i(X\times Y_i).$$ It doesn't seem like this should be the case since on the right hand side, the open set from $X$ could be taken to be arbitrary small as we go over the indices, although I might be mistaken. Could this hold if there are finite number of $Y_i$?

Answer 1

Yes, the homeomorphism holds for any index set $I$. We map $(x,(y,i))$ to $((x,y),i)$, essentially.

Bijection is trivial, both ways continuous likewise. I don't see issues like you describe.