In May's Algebraic topology book, he states the following:
Proposition: If $X$ is compactly generated and $\pi:X\to Y$ is a quotient map, then $Y$ is compactly generated $\iff$ $(\pi\times\pi)^{-1}(\Delta Y)$ is closed in $X\times X$.
Here's some notation that May uses: For topological spaces $Z$, $Z'$ We write $Z\times_c Z'$ as the product of $Z$ and $Z'$ with the usual topology, and $Z\times Z'$ as the $k$-ification of $Z\times_c Z'$.
Note that compactly generated (CG) means being weak Hausdorff (WH) and a $k$-space, $k$-space meaning all compactly closed sets are closed.
The $\Rightarrow$ implication is okay, Y being CG implies $\Delta Y$ is closed in $Y\times Y$, implying $(\pi\times\pi)^{-1}(\Delta Y)$ is closed in $X\times X$.
I'm partially done with the $\Leftarrow$ implication. I know how to prove that $Y$ is $k$-closed. Suppose that $A$ is a compactly closed set. It suffices to prove that $\pi^{-1}(A)$ is closed. Since $X$ is CG, it suffices to prove that $\pi^{-1}(A)$ is compactly closed. Let $g:K\to X$ be a map from a compact space $K$. Then $g^{-1}(\pi^{-1}(A))=(\pi\circ g)^{-1}(A)$, which is closed. Therefore, $\pi^{-1}(A)$ is compactly closed.
I'm having a hard time proving that $Y$ is WH in the $\Leftarrow$ implication. I would appreciate any hints!
Best Answer
Using proposition 2.20 in the linked notes in the comments, we can prove that $\pi\times \pi$ is a quotient map. Thus, $\Delta Y=(\pi\times \pi)((\pi\times \pi)^{-1}\Delta Y)$ is a closed set in $Y\times Y$. This finishes proving that $Y$ is WH.