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Reverse Mathematics and Π12 Comprehension

Published online by Cambridge University Press:  15 January 2014

Carl Mummert  and
Stephen G. Simpson
Carl Mummert
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, State College, PA, 16802, USA, E-mail: mummert@math.psu.edu, URL: http://www.math.psu.edu/mummert/, E-mail: simpson@math.psu.edu, URL: http://www.math.psu.edu/simpson/
Stephen G. Simpson
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, State College, PA, 16802, USA, E-mail: mummert@math.psu.edu, URL: http://www.math.psu.edu/mummert/, E-mail: simpson@math.psu.edu, URL: http://www.math.psu.edu/simpson/
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Abstract

We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 11 , Issue 4 , December 2005 , pp. 526 - 533
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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