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Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ such that $f$ is a local homeomorphism if and only if $\tilde f$ has property $P$ (and I don't want this to be a tautology).

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Why do you think that it makes a difference if you restrict yourself to compactly generated spaces? – Dominic van der Zypen Dec 12 '14 at 9:49
I need $Y^X$ to exist, so I was restricting to a Cartesian closed subcategory. in $Top$, this is only possible if $X$ is core-compact. – David Carchedi Dec 12 '14 at 17:53
Well, the function space $X^Y$ can be defined for any topological spaces, inheriting the topology from the product topology.. – Dominic van der Zypen Dec 13 '14 at 13:15
OK, fair enough. I follow the category-theorist convention of only writing $X^Y$ for a genuine internal-hom, that is, something satisfying the correct universal property, but at any rate, there is no risk of confusion in my question, since I restrict to compactly generated spaces, which are Cartesian-closed. – David Carchedi Dec 15 '14 at 6:08

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