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Association for Symbolic Logic
Journal of Symbolic Logic
Original Item ID
A first order representation (f.o.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions "one f.o.r. is at least as expressive as another relative to a class of spaces" and "one class of spaces is definable in another relative to an f.o.r.", and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positive universal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting.
Bankston, Paul, "Expressive Power in First Order Topology" (1984). Mathematics, Statistics and Computer Science Faculty Research and Publications. 131.