Let X and Y be topological spaces, and let $X\times Y$ be the corresponding product space. Define the projection functions
$$p_X:X\times Y \rightarrow X \mbox{ and } p_Y:X\times Y \rightarrow Y$$
by $p_X(x,y)=x$ and $p_Y(x,y)=y$. Prove that $p_X$ and $p_Y$ are continuous.
So it's clear to me that if $x$ is open in $X$, then $p_X^{-1}(x)=(x,y)$ but I don't understand how we know that $(x,y)$ is open just because $x$ is open. Isn't the basis for a product space $B:=\{U×V∣U⊆X \mbox{ open },V⊆Y \mbox{ open }\}?$ How do we say $(x,y)$ is open in $X\times Y$ without knowing whether $y$ is open in $Y$?
For any open set $U\subset X$, the preimage $p_X^{-1}(U)=U\times Y$ is open in $X\times Y$ since $U$ is open in $X$ and $Y$ is open in $Y$ (this is true of any topology on $Y$, just from the definition of a topology).
For any open set $V\subset Y$, the preimage $p_Y^{-1}(V)=X\times V$ is open in $X\times Y$ since $X$ is open in $X$ and $V$ is open in $Y$.