In his Notes on Set Theory (p. 44) Moschovakis defines:
A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$.
But even when $S$ is allowed to be something else than a set, this is hardly a definition like
A graph is a pair $(V,E)$ with $E \subseteq V \times V$
or
A topological space is a pair $(X,T)$ with $T \subseteq \mathcal{P}(X)$ ...
because nothing is imposed on the relationship between $A$ and $S$. The "definition" looks like the definition of (the notion of) an ordered pair, and at most it's a necessary condition for structured sets.
But how can one require that the frame $S$ is not an arbitrary object but has to do with the underlying set $A$ (to be able to endow some structure on it)?
Looking at the typical examples of algebraic and relational structures and of topological spaces one might tend to require that $S$ is a tupel $(S_1,\dots,S_n)$ with $\mathsf{ZFC}\vdash S_i \in X_i(A)$ where $X_i(A)$ is a set constructed over $A$ solely by iterating the set operations $\mathcal{P}$ and $\times$. (This seems to be the Bourbakian way of echelons.)
But how would this restriction be justified? Why not requiring that $\mathsf{ZFC}\vdash \phi(S_i,A)$ for an arbitrary formula $\phi(X,Y)$?
OK, this would not do the job, because $X = X \wedge Y = Y$ is such a formula but doesn't induce a genuine relationship between $X$ and $Y$ like e.g $X \subseteq Y \times Y$ does.
Can we characterize a truely general family of formulas $\phi(S_i,A)$ that do induce such a relationsship – beyond those of the form $S_i \in X_i(A)$?
Suppose we can: Can we make this characterization into a first-order definition of "structured set"?
The second question has to do with the fact that one cannot give a first-order definition of "definable sets" but e.g. of "constructible sets".