  Various topological concepts are often involved in the research of
mathematical logic, and almost all of these concepts can be regarded as
developing from the Stone representation theorem. In the Stone representation
theorem, a Boolean algebra is represented as the algebra of the clopen sets of
a Stone space. And based on this, a natural connection is established between
the structure of Stone space and the semantics of propositional logic. In other
words, models of a propositional theory are represented as points in a Stone
space. This enables us to use the concepts of topology to describe many facts
in logic. In this paper, we do the same thing for the first-order logic. That
is, we organize the basic objects of semantics of first-order logic, such as
theories, models, elementary embeddings, and so on, into a kind of topological
structure defined abstractly. To be precise, this kind of structure is a kind
of enriched-topological space which we call cylindric space in this paper.
Furthermore, based on this topological representation of semantics of
first-order logic, we systematically introduce a method of point-set topology
into the research of model theory. We demonstrate the great advantages of this
topological method with an example and provide a general discussion of its
features, advantages, and connection to the type space.
