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Expressive power in first order topology

Published online by Cambridge University Press:  12 March 2014

Paul Bankston
Paul Bankston*
Affiliation:
Marquette University, Milwaukee, Wisconsin 53233
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Abstract

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.

Keywords

first order topologyfirst order representationszero-setsrings of continuous functionselementary equivalence.
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 478 - 487
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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