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.

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).

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