  We survey discrete and continuous model-theoretic notions which have
important connections to general topology. We present a self-contained
exposition of several interactions between continuous logic and $C_p$-theory
which have applications to a classification problem involving Banach spaces not
including $c_0$ or $l^p$, following recent results obtained by P. Casazza and
J. Iovino for compact continuous logics. Using $C_p$-theoretic results
involving Grothendieck spaces and double limit conditions, we extend their
results to a broader family of logics, namely those with a first countable
weakly Grothendieck space of types. We pose $C_p$-theoretic problems which have
model-theoretic implications.
