On embedding separable spaces C ( L ) in arbitrary spaces C ( K )

Supplementing and expanding classical results, for compact spaces K and L, L metric, and their Banach spaces $C\left(L\right)$ and $C\left(K\right)$ of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of $C\left(L\right)$ into $C\left(K\right)$ . In particular, we show that if the embedded space $C\left(L\right)$ is separable, then the classical theorems of Holsztyński and Gordon become equivalences. We also obtain new results describing the relative cellularities of the perfect kernel of a given compact space K and of the Cantor-Bendixson derived sets of K of countable order in terms of the presence of isometric copies of specific spaces $C\left(L\right)$ inside $C\left(K\right)$ .