Function spaces on separable compact lines

Kacper Kucharski

Function spaces on separable compact lines

General Topology 2026-07-02 v1 Functional Analysis

Authors: Kacper Kucharski

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.

Cite

@article{arxiv.2607.02713,
  title  = {Function spaces on separable compact lines},
  author = {Kacper Kucharski},
  journal= {arXiv preprint arXiv:2607.02713},
  year   = {2026}
}