with parameters from $M$. By $S_{G,\Delta}(M)$ we denote the compact space of complete $\Delta$-types over $M$ concentrating on $G$, equivalently, the space of ultrafilters of relatively $\Delta$-definable over $M$ subsets of $G$. In the case when $\Delta$ is the collection of all formulas in the variable $x$ of the same sort as $G$ and arbitrary parametric variables, $\Delta$-formulas over $M$ are just all formulas over $M$ in the variable $x$, and $S_{G,\Delta}(M)=S_G(M)$.

Sometimes we will be using $\Delta$-types over sets of parameters other than $M$. In such situations, all the definitions of “$\Delta$-objects” are analogous, except we take the convention that by a $\Delta$-formula over $A$ we mean a formula over $\operatorname{{dcl}}(A)$ which is equivalent to a Boolean combination of instances of formulas from $\Delta$ with parameters from $\operatorname{{dcl}}(A)$ (we do it in order to avoid writing “$\operatorname{{dcl}}$” many times in the paper). Another option is to define $\Delta$-formulas over $A$ as those formulas over $A$ which are equivalent to Boolean combinations of instances of formulas from $\Delta$ with arbitrary parameters (but then some statements would be slightly less general).

We naturally extend the definition of a definable function from $G$ to a compact space $C$ to a $\Delta$-definable context.

