# Let $f:X \to Y$ be a quotient map and $X$ and $Y$ hausdorff spaces, where $X$ is compactly generated. Why is $Y$ then compactly generated?

This question is related to this one:

Any quotient of a compactly generated space is compactly generated

The difference is that I assume the spaces to be hausdorff additionally. So I was wondering, if there is an easier way to prove the statement then without using this machinery then. Basically, a simple proof fails because I cannot say that $f^{-1} (K)$ is compact for $K \subset Y$ compact in my opinion. Do you have any hints or suggestions about that problem?

• But I only assume $X$ to be compactly generated. Are compactly generated hausdorff spaces compact in general? – MPB94 Nov 29 '17 at 13:40
• Oops, sorry, missed that point. I should delete that comment. – Lee Mosher Nov 29 '17 at 13:41