While looking at a proposition from Hatcher AT:
Proposition A.15. If $X$ is a compactly generated Hausdorff space and $Y$ is locally compact, then the product topology on X×Y is compactly generated.
Is it necessary for $X$ to be Hausdorff in order for $X \times Y$ to be compactly generated?
Below I give an argument why the condition is not needed, but this theory is new to me, and I am extremely unsure about it's validity.
Hatcher uses the condition of Hausdorffness of $X$, in the fact that a compact Hausdorff space is locally compact, which I think it can be avoided by my below argument:
Argument: We want to prove $f:X\times Y \rightarrow (X\times Y)_c$ is continuous. Since $Y$ is locally compact is enough to prove that for any $K\subset X$, compact, the function $f:K\times Y \rightarrow (X\times Y)_c$ is continuous.
So fix such a $K\subset X$ compact. We know that $f:K\times C \rightarrow (X\times Y)_c$ is continous for any compact $C\subset Y$. For each $y\in Y$, we pick a compact neighborhood $C_y$. Since $f$ is continuous on $K \times C_y$, it must be that $f$ is continuous on $K\times int(C_y)$, which is an open subset of $K\times C$. Therefore $f$ is continous on \begin{equation} \bigcup_{y\in Y} K\times int(C_y)=K \times Y, \end{equation}
the idea is that $f$ is continuous on a cover of open sets of $K \times Y$, therefore is continuous on $K \times Y$.