# If $C(K)$ is isomorphic to $C(H)$, are $K$ and $H$ homeomorphic?

Let $$K,H$$ be compact Hausdorff spaces. The Banach Stone theorem says that $$K$$ and $$H$$ are homeomorphic $$\iff$$ $$C(K)$$ and $$C(H)$$ are isometrically isomorphic. Is it true that if $$C(K)$$ and $$C(H)$$ are just isomoprhic, that $$K$$ is homeomorphic to $$H$$?

If $$K$$ and $$H$$ are both uncountable, then by Miljutin theorem (see theorem 4.4.8 in Topics in Banach space theory by Fernando Albiac, Nigel J. Kalton) $$C(K)$$ and $$C(H)$$ are isomorphic.
Thus there is an abundance of counterexamples: take any two uncountable topological spaces that are not homeomorphic, say $$K=[0,1]$$, $$H=[0,1]\cup\{2\}$$.