Let $K$ be a compact Hausdorff infinite topological space and $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$ with sup norm. It is known that $c\_0$ is complemented in $C(K)$ for $K$ that contains infinite convergent sequence. I would appreciate if somebody let me know an example of $K$ without infinite convergent sequences such that $C(K)$ contains a complemented subspace isomorphic to $c\_0$. 



Compact space $K$ without convergent sequence while $c_0$ is complemented in $C(K)$
Take $K = \beta N \times \beta N$, the square of the Čech--Stone compactification of the integers. The space of continuous functions on that space is not a Grothendieck spaces. A space of continuous functions on a compact space is Grothendieck if and only if it doesn't contain complemented copies of $c\_0$.


More generally you may take any infinite $K = L \times L$, such that $C(L)$ is a Grothendieck space.


