Supplementing and expanding classical results, for compact spaces K and L, L metric, and their Banach spaces \({\mathcal {C}}(L)\) and \({\mathcal {C}}(K)\) of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of \({\mathcal {C}}(L)\) into \({\mathcal {C}}(K)\) . In particular, we show that if the embedded space \({\mathcal {C}}(L)\) 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 \({\mathcal {C}}(L)\) inside \({\mathcal {C}}(K)\) .
