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

  • $\begingroup$ Yes, if the exponential law holds for a space (in the compact-open topology), it implies it's locally compact Hausdorff. So it's indeed necessary; this has been known since the 1950's I think. $\endgroup$ – Henno Brandsma Feb 28 at 11:45

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 $\mathbb{N}^\mathbb{N}$.


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