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?

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

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