Homeomorphism between $X\times Y$ with product topology and $\mathbb{R}$ implies either X or Y are singletons?

Question

If you have a homeomorphism $f:X\times Y \rightarrow \mathbb{R}\:$, where $X\times Y$ is using the product topology and $\mathbb{R}$ the standard topology, does this imply that either $\#X = 1$ or $\#Y = 1$ ? If so, how would you prove that?