Let $X,X',Y,Y'$ be compact Hausdorff spaces.
Consider the spaces of continuous maps $C(X,Y)$ and $C(X',Y')$ with the topology of uniform convergence.
Is it true that $C(X,Y)$ and $C(X',Y')$ are homeomorphic?
A candidate homeomorphism:
Call the homeomorphisms $h_X : X \to X'$ and $h_Y : Y \to Y'$. Define $H: C(X,Y) \to C(X',Y')$ by $H(f) = h_Y \circ f \circ h_X^{-1}$.
$H^{-1} : C(X',Y') \to C(X,Y)$ is $H^{-1} (g) = h_Y^{-1} \circ g \circ h_X$.
Note that $H^{-1} \circ H (f) = f$.
We know that a continuous bijection between compact spaces is a homeomorphism.
Thus it remains to show the continuity of $H$, which I fail to do.
Many thanks.