$C_p$-Theory for Model Theorists

TL;DR

This paper explores applications of $C_p$-theory, a branch of topology, to model theory and continuous logic, providing new results and generalizations relevant to definability of Banach spaces.

Contribution

It introduces $C_p$-theoretic techniques to model theorists, generalizes previous results on Banach space definability, and bridges topology with continuous logic.

Findings

Generalized results on the definability of Banach spaces including $c_0$ and $\\ell^p$

Provided self-contained $C_p$-theoretic proofs for model theorists

Extended prior work of Casazza, Iovino, Odell, and Gowers

Abstract

We present applications of $C_p$-theory, the branch of general topology concerned with spaces of real-valued continuous functions, to model theory, mostly in the context of continuous logics. We include $C_p$-theoretic results and proofs in a self-contained way for model theorists who are not familiar with the techniques of this field. We further generalize some results of Casazza and Iovino, and of the authors, involving the definability of Banach spaces including isomorphic copies of $c_0$ or $\ell^p$, after a problem posed by Odell and Gowers.