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.
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