On embedding separable spaces $\mathcal{C}(L)$ in arbitrary spaces $\mathcal{C}(K)$

TL;DR

This paper characterizes when separable spaces of continuous functions on compact spaces can be embedded into others, revealing new equivalences and relations involving cellularities and derived sets.

Contribution

It provides new characterizations of isometric and isomorphic embeddings of $ ext{C}(L)$ into $ ext{C}(K)$, especially for separable spaces, expanding classical results.

Findings

Equivalence of classical theorems for separable spaces

Characterizations of embeddings based on cellularities

Relations between derived sets and embeddings

Abstract

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 Holszty\'{n}ski 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 $\mathcal{C}(K)$.