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Let $K$ be compact Hausdorff topological space, $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$, endowed with supremum norm. It is known that if $K$ contains convergent sequence then $c_0$ is complemented in $C(K)$. I would appreciate if somebody let me know an example of a compact Hausdorff $K$ without convergent infinite sequences such that $c_0$ is complemented in $C(K)$.

Aligomez
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This question has been answered by Tomek Kania at mathoverflow. A example given by he is $\beta \mathbb{N} \times \beta \mathbb{N}$. Thanks to all who try to answer this question here.

Aligomez
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  • That striled me as curious - $C(K\times K)$ contains a complemented $c_0$ while $C(K)$ does not. (You might provide a link to make it easier to find...) – David C. Ullrich Jul 24 '19 at 22:45
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    this is the link: https://mathoverflow.net/questions/336827/compact-space-k-without-convergent-sequence-while-c-0-is-complemented-in-c#comment845539_336830. – Aligomez Oct 11 '19 at 20:25