$\coprod_{x\in X} (\{x\}\times Y)$ and $X\times Y$ are homeomorphic

Question

Let $X$ and $Y$ be topological spaces. Then is it true that $X\times Y$ is homeomorphic to $\coprod_{x\in X} (\{x\}\times Y)$? Clearly these two can be identified as sets, but I'm not sure that these are homeomorphic.

Answer 1

They will be homeomorphic if $X$ is discrete, but not in general (just let $Y$ be a one-point space).

Answer 2

Hint: A set $A \subseteq \coprod_{x \in X} \{x\} \times Y$ is open iff $A$ is open in $\{x\} \times Y \cong Y$ for all $x \in X$.

This shows that both topologies are not homeomorphic in general, as the openness of a set in the coproduct does not depend on the first factor $X$ in some way.

Now, make this formal (by making an explicit counterexample).