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Various topological concepts are often involved in the research ofmathematical logic, and almost all of these concepts can be regarded asdeveloping from the Stone representation theorem. In the Stone representationtheorem, a Boolean algebra is represented as the algebra of the clopen sets ofa Stone space. And based on this, a natural connection is established betweenthe structure of Stone space and the semantics of propositional logic. In otherwords, models of a propositional theory are represented as points in a Stonespace. This enables us to use the concepts of topology to describe many factsin logic. In this paper, we do the same thing for the first-order logic. Thatis, we organize the basic objects of semantics of first-order logic, such astheories, models, elementary embeddings, and so on, into a kind of topologicalstructure defined abstractly. To be precise, this kind of structure is a kindof enriched-topological space which we call cylindric space in this paper.Furthermore, based on this topological representation of semantics offirst-order logic, we systematically introduce a method of point-set topologyinto the research of model theory. We demonstrate the great advantages of thistopological method with an example and provide a general discussion of itsfeatures, advantages, and connection to the type space.

A Topological Representation of Semantics of First-order Logic and Its Application as a Method in Model Theory