TITLE: On $C(K)$ spaces embeddable into the Banach space $c_0$
QUESTION [8 upvotes]: Problem 1. Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$.

Since $c_0$ has separable dual, such $K$ must me countable. So, we can make Problem 1 more precise:

Problem 2. Is it true that for every compact countable space $K$ the Banach space $C(K)$ is isomorphic to a subspace of $c_0$?

Another possible option:

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 $c_0$?

REPLY [13 votes]: 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.