Homeomorphism between function spaces

Question

I was reading proof for $Map(X\times Y, Z) \cong Map(X, Map(Y,Z))$. As a corollary 7.14, pg4, author wrote,

Corollary: $Map_0(X \wedge Y, Z) \cong Map_0(X, Map_0(Y,Z))$ in Top$^0$.

How does this follow?

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Suppose I want to past on this result to the category of CGWH spaces. How does one do this rigorously?

Answer 1

Since pointed maps $X \wedge Y \to Z$ are precisely pointed maps $X \times Y \to Z$ that send the wedge sum of $X$ and $Y$ to the base point in $Z$, by the adjunction we know that such maps correspond to certain maps $X \to Maps(Y,Z)$. Can you figure out which these are?

These adjunctions work for all reasonable spaces.

Answer 2

I don't think that $Map_0(X \wedge Y, Z) \cong Map_0(X, Map_0(Y,Z))$ is a direct corollary of $Map(X\times Y, Z) \cong Map(X, Map(Y,Z))$. You can easily verify that there is a canonical bijection, but to show that it is a homeomorphism needs extra arguments.

See for example Section 2.4 of

tom Dieck, Tammo. Algebraic topology. Vol. 8. European Mathematical Society, 2008.