I'm trying to prove that if $X$ is compactly generated and $Y$ is T2 (Hausdorff) and locally compact then $X\times Y$ is compactly generated.
First it is clear that since both $X$ and $Y$ are T2 then $X\times Y$ is also T2 and so the first condition for "compactly generated" is fulfilled.
For the next condition, we need to show that: $$\forall B\subseteq X\times Y,\,\left[B\in Open(X\times Y)\Longleftrightarrow B\cap K\in Open(K)\forall K\in Compact(X\times Y) \right]$$
So let $B\subseteq X\times Y$ be given.
The direction $\Longrightarrow$ follows immediately due to the definition of the subspace topology, so there is nothing to prove.
For the $\Longleftarrow$ direction, assume $B\cap K\in Open(K)\forall K\in Compact(X\times Y)$, and the goal is to show $B\in Open(X\times Y)$.
My idea was to take an arbitrary point $(x,y)\in B$ and try to find $U\times V \in Open(X)\times Open(Y)$ such that $(x,y)\in U\times V \subseteq B$.
Because $Y$ is locally compact, $\exists K_y \in Compact(Y)$ such that $\exists V_y \in Open(Y)$ such that $y\in V_y \subseteq K_y$.
However, now I get stuck, because I don't know which compact set $K_x$ of $X$ to find so that $x\in K_x$ and $(x,y)\in K_x \times K_y$.
Because there is no obvious choice for a compact set, I'm not sure how to employ the input data.