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We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivialtwisted sum with $Y$. We present a characterization of such space $Y$ in termsof properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under thecontinuum hypothesis $c_0$ has a nontrivial twisted sum with every space of theform $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistentpositive solution to a problem posed by Cabello, Castillo, Kalton and Yost.

Twisted sums of $c_0$ and $C(K)$-spaces: A solution to the CCKY problem