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Let $T$ be the $\theta$-type Calder\'on-Zgymund operator with Dini condition.In this paper, we prove that for $b\in {\rm CMO}(\mathbb R^n)$, the commutatorgenerated by $T$ with $b$ and the corresponding maximal commutator, are bothcompact operators on $L^{p}(\omega)$ spaces, where $\omega$ be the Muchenhoupt$A_p$ weight function and $1<p<\infty$.

The compactness of commutators of Calder\'on-Zgymund operators with Dini condition