Is $U\times\{y\}$ ever open in $X\times Y$, here $X$ and $Y$ are Banach spaces?

Question

Let $U$ be an open set in a Banach space $X$, let $Y$ be another Banach space, and let $y\in Y$. Is it ever the case that \begin{align*} U\times\{y\} \end{align*} is an open set in $X\times Y$?

Answer 1

This would only be the case if $\{y\}$ is open in $Y$, which would imply that $Y = \{0\}$.

Answer 2

"Is it always the case that"? No, why should it? Let $X = Y = \Bbb R$, let $U = (0, 1)\subseteq \Bbb R$, and let $y = 0$. The (open) line segment from $(0, 0)$ to $(1, 0)$ is not open in $\Bbb R^2$.

Whenever you have an idea of some property, you should always check it out in the most familiar settings (in the case of vector spaces, and much of topology and analysis, it's most likely $\Bbb R$ or $\Bbb R^2$, and in algebra it's usually $\Bbb Z$, possibly modulo some small number). Only if the property does work in that case should you ask the question of whether it works for all of them.

Now that you've edited your question to read "Is it ever the case that", I can answer that one as well. It is the case only if $\{y\} \subseteq Y$ is an open set. That only happens if $Y$ is a zero-dimensional banach space.