# Hausdorff compactly generated is locally compact?

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.

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.
You can take any non-locally compact metric space, for some familiar examples. E.g., the Baire space $$\mathbb{N}^\mathbb{N}$$.