I was trying to prove the "theorem of exponential correspondence" and "exponential law" with $X,Y,Z$ compactly generated and Hausdorff.
According to Hedwin H.Spanier-Algebraic Topology and Tammo tom Dieck-Algebraic topology we have that locally compact seems necessary in order to prove the statement.
The true hyphotesis given on the spaces are Hausdorff locally compact, but as far as I can see, compactly generated and Hausdorff doesn't imply locally compact, am I right ? Are my hypothesis wrong ? Are there any counterexample.
Thanks in advance
Best Answer
A simple example would be $\Bbb Q$, endowed with its usual topology. It is metrizable, and therefore compactly generated. But it is not locally compact.