Name of topology analogous to Zariski topology for an arbitrary model of a first-order theory In the Zariski topology, a set of points in $A^n$, $w$, is closed by definition when there exists a set of polynomials in $n$ variables, $S$, such that the following holds.
$$ \forall u \in A^n \mathop. \bigg( u \in w \iff (\forall f \in S \mathop. f(u) = 0) \bigg) $$
By squinting, we can come up with a topology with a similar definition that works for an arbitrary model of a first-order theory and convince ourselves that it's really a topology.
I'm wondering what this topology's name is (if it has one) and whether it is at all useful for classifying models in some way.
Let $L$ be a language with constant, function, and relation symbols.
Let $\theta$ be a set of $L$-sentences.
Let $M$ be our model, $M \models \theta$.
Let $V$ be a set of well-formed formulas where each well-formed formula has exactly one free variable $x$. We can assume that the free variable is named $x$ without losing any generality.
We define $w$ to be a closed set when the following holds:
$$ \forall u \in M_D \mathop. \bigg( u \in w \iff (\forall v \in V \mathop. v[x:=u] \;\;\text{is true in $M$}) \bigg) $$
We can show that this thing is really a topology.
Let $F$ be a family of sets of wff's. The subset of $M_D$ associated with $\cup F$ is closed, therefore the closed sets of our topology are closed under arbitrary intersection.
To show closure under finite union, it suffices to show closure under binary unions.
Suppose we have two sets of wffs with one free variable $A$ and $B$. Further suppose that in each wff in $A$ and each wff in $B$ the free variable is named $x$.
Define $C$ as follows:
$$ C \stackrel{\text{def}}{=\!=} \{ a \lor b \mathop. a \in A \land b \in B \} $$
In other words, we take every pair of wffs where the left formula comes form $A$ and the right formula comes from $B$ and join them together with a $\lor$.
The empty set is given by the formula $\exists x \mathop. x \neq x$.
The whole domain of the model is given by the formula $\exists x \mathop. x = x$.
 A: I'm not sure what you mean by $M_D$. If you mean the underlying set of $M$ and you consider all formulas with parameters in $M$, then this is simply the discrete topology.
I believe the topology you are trying to describe is simply the restriction of the usual Stone topology in the type space $S_x(A)$ (for a fixed $A\subseteq M$, perhaps simply $A=\emptyset$) to the subspace of types of elements of $M$.
I don't think it has any widely-used name. If I would refer to it, I would just describe it as I did in the preceding paragaph. If I had to name it, I would just (abusively) call it the Stone topology (in $M$ over $A$). The Stone topology on the space of types is definitely useful in model theory, since it allows us to talk about definable phenomena in essentially purely topological terms. This particular restriction --- I'm not so sure about.

Edit: In hindsight, this topology will certainly not reproduce the Zariski topology in the case of algebraically closed fields. To do that, you need to consider some refinements: instead of all formulas, you can only consider formulas of particular form, e.g. quantifier-free positive formulas (formulas with no quantifiers nor negations). (Note: thanks to Alex Kruckman to pointing out an erreneous remark about pp-formulas being equivalent to positive quantifier-free formulas.)
Then the resulting topology on $M$ is not discrete. Still, even in this case, I think it would be more natural to consider the topology on the space of complete (qf positive) types. I think this is similar to how in algebraic geometry an affine scheme is not completely captured by its underlying Zariski topological space. (Although as noted by Alex in the comments, the closer analogy is the topology on the spectrum of the coordinate ring of an affine variety.)
