Open sets in product topology
For any two topological spaces $X$ and $Y$, consider $X \times Y$. Is it always true that open sets in $X \times Y$ are of the forms $U \times V$ where $U$ is open in $X$ and $V$ is open in $Y$?
I think is no. Consider $\mathbb{R}^2$. Note that open ball is an open set in $\mathbb{R}^2$ but it cannot be obtained from the product of two open intervals.
Nice done. The forms $U\times V$ is a base for the topology. That means that every open set in $X\times Y$ is a union of elements of the form $U\times V$.