Say $h:\prod X_{\alpha}\rightarrow \prod X_{\alpha}$ and $\space h(x)=(h_{\alpha}(x_{\alpha}))$, $U = \prod U_{\alpha}$ is an open set in $\prod X_{\alpha}$.
It's $h(U) = \prod (h_{\alpha}(U_{\alpha}))$ true? What if $U$ is only open in $\prod X_{\alpha}$
If it's true, why? What if $h(U)$ is a union of different cubes (or square, a visualization that helps me understand for a product of 2 $X_{\alpha}$)?
If it's wrong, is this true if $U$ is a basis? $h(U)$ can still be a union of different cubes even if $U$ itself is a cube right?
I'm really confused, appreciate it!