Generalized geometric theories and set-generated classes

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is that the class of models of a generalized geometric theory is set-generated. Here a class X of subsets of a set is set-generated if there exists a subset G of X such that for each α ∈ X and finitely enumerable subset τ of α there exists a subset β ∈ G such that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo-Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable in CZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.