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Let $C_0(K, X)$ denote the space of all continuous $X$-valued functionsdefined on the locally compact Hausdorff space $K$ which vanish at infinity,provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K,X)$ by simply $C_0(K)$. In this paper we prove that for locally compactHausdorff spaces $K$ and $L$ and for Banach space $X$ containing no copy of$c_0$, if there is a isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ whereeither $X$ is separable or $X^*$ has the Radon-Nikod\'ym property, then either$K$ is finite or $|K|\leq |L|$. As a consequence of this result, if there is aisomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where $X$ contains no copy of$c_0$ and $L$ is scattered, then $K$ must be scattered.

On embeddings of $C_0(K)$ spaces into $C_0(L,X)$ spaces