We study interactions between general topology and the model theory of real-valued logic. This thesis includes both applications of topological ideas to obtain results in pure model theory, and a model-theoretic approach to the study of compacta via their rings of continuous functions viewed as metric structures.
We introduce an infinitary real-valued extension of first-order continuous logic for metric structures which is analogous to the discrete logic $\mathcal{L}_{\omega_1, \omega}$, and use topological methods to develop the model theory of this new logic. Our logic differs from previous infinitary logics for metric structures in that we allow the creation of formulas $\inf_n \phi_n$ and $\sup_n \phi_n$ for all countable sequences $(\phi_n)_{n < \omega}$ of formulas, while previous versions required that the resulting formulas define uniformly continuous functions. Our more general context allows us to axiomatize several important classes of structures from functional analysis which are not captured by previous logics for metric structures. We give a topological proof of an omitting types theorem for this logic, which gives a common generalization of the omitting types theorems of Henson and Keisler. Consequently, we obtain a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces. We show that continuous functions on separable metric structures are definable in our $\mathcal{L}_{\omega_1, \omega}$ if and only if they are automorphism invariant, and show that no such result holds for the other logics for metric structures.
The second part of this thesis develops the model theory of the C*-algebras $C(X)$, for $X$ a compact Hausdorff space. We describe all complete theories of these algebras for $X$ a compact $0$-dimensional space. We show that the complete theories of $C(X)$ (for $X$ of any dimension) having quantifier elimination are exactly the theories of $\mathbb{C}$, $\mathbb{C}^2$, and $C(2^{\mathbb{N}})$, and that if the theory of $C(X)$ is model complete and $X$ is connected then $X$ is co-elementarily equivalent to the pseudoarc. We use model-theoretic forcing to answer a question of P. Bankston by showing that the pseudoarc is a co-existentially closed continuum. Finally, we study how model-theoretic saturation of $C(X)$ relates to topological properties of $X$.