Function spaces on separable compact lines

Kacper Kucharski

paper

Function spaces on separable compact lines

arXiv:2607.02713

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 $I ¨ _{1}$ , under the assumption of Baumgartner's Axiom $BA$ . 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^{I ¨} $ neither of whose spaces of continuous functions, $C_{p} (K)$ and $C_{w} (K)$ , is homeomorphic to its square.