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**1 - 2**of**2**### Generalized geometric theories and set-generated classes

, 2012

"... 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. ..."

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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.

### Under consideration for publication in Math. Struct. in Comp. Science Spatiality for formal topologies

, 2006

"... We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples. 1. ..."

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We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples. 1.