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

## TL;DR

This paper explores when spaces of continuous functions on one compact space can be embedded into another, focusing on separable spaces and extending classical theorems.

## Contribution

The paper provides new characterizations for isometric and isomorphic embeddings of separable function spaces into arbitrary ones.

## Key findings

- If the embedded space is separable, Holsztyński and Gordon's theorems become equivalences.
- New results link the structure of compact spaces to the presence of isometric copies of specific function spaces.
- The relative cellularities of derived sets are described in terms of embedding properties.

## Abstract

Supplementing and expanding classical results, for compact spaces K and L, L metric, and their Banach spaces \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(L)$$\end{document}C(L) and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(K)$$\end{document}C(K) of continuous real-valued functions, we provide several characterizations of the existence of isometric, resp. isomorphic, embeddings of \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(L)$$\end{document}C(L) into \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(K)$$\end{document}C(K). In particular, we show that if the embedded space \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(L)$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(L)$$\end{document}C(L) inside \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}(K)$$\end{document}C(K).