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> **Problem 1.** Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c\_0$.
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Since $c\_0$ has separable dual, such $K$ must me countable. So, we can make Problem 1 more precise:



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> **Problem 2.** Is it true that for every compact countable space $K$ the Banach space $C(K)$ is isomorphic to a subspace of $c\_0$?
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Another possible option:



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> **Problem 3.** Let $K$ be a compact Hausdorff space. Is it true that the Banach space $C(K)$ is isomorphic to $c\_0$ if $C(K)$ is isomorphic to a subspace of $c\_0$?
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On $C(K)$ spaces embeddable into the Banach space $c_0$
The Szlenk index is the answer. 


A space $C(K)$, where $K$ is infinite compact Hausdorff space, is embeddable into $c\_0$ if and only if $K$ is homeomorphic to an ordinal below $\omega^\omega$ and if this is the case (and $K$ is infinite) the space itself is isomorphic to $c\_0$.


So the answer to problem 2 is no however the answer to problem 3 is yes.


For details see Rosenthal's chapter in the Handbook of Banach spaces.


