Function spaces on separable compact lines

Kacper Kucharski

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arXiv · 2607.02713

Function spaces on separable compact lines

Kacper Kucharski

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Abstract

In this paper, we provide a complete isomorphism classification of the spaces $C_p(K)$ of real-valued continuous functions endowed with the topology of pointwise convergence for separable compact lines $K$ of weight $\omega_1$, under the assumption of Baumgartner's Axiom $\mathsf{BA}$. More specifically, we show that, up to linear homeomorphism, there are exactly two function spaces $C_p(K)$ for such $K$. We also construct an example of a separable compact line $K$ of weight $2^{\omega}$ neither of whose spaces of continuous functions, $C_p(K)$ and $C_w(K)$, is homeomorphic to its square.

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Kacper Kucharski. 2026-07-02. Function spaces on separable compact lines. https://arxiv.org/abs/2607.02713

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