  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.
