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Let $X$ be a compact Hausdorff topological space whose convergent sequences are eventually constant. Is there a description of such spaces. How ''far'' these spaces from Stonean ones?

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    $\begingroup$ $\beta \mathbb{N}$ (the Stone-Čech compactification of the discrete space $\mathbb{N}$) has this property, but is also extremally disconnected, hence Stonean. $\endgroup$
    – user642796
    Aug 15 '15 at 23:19
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Norbert, I don't think that there is a reasonable description of such spaces. Indeed, they properly contain spaces $K$ for which the Banach space $C(K)$ is Grothendieck and this class is far from being fully delineated. For example, there are connected, compact spaces $K$ for which $C(K)$ is indecomposable, hence Grothendieck.

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