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December 2005 Reverse mathematics and $\Pi_{2}^{1}$ comprehension
Carl Mummert, Stephen G. Simpson
Bull. Symbolic Logic 11(4): 526-533 (December 2005). DOI: 10.2178/bsl/1130335208

Abstract

We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to $\Pi_{2}^{1}$ comprehension. An MF space is defined to be a topological space of the form MF($P$) with the topology generated by {$N_p \,| \,p \in P$}. Here $P$ is a poset, MF($P$) is the set of maximal filters on $P$, and $N_p =${$F \in \rm{MF}(P)| p \in F$}. 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 $\mathsf{ACA}_0$ of second order arithmetic. One can prove within $\mathsf{ACA}_0$ 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 $\Pi_{2}^{1}-\mathsf{CA}_0$ The equivalence is proved in the weaker system $\Pi_{1}^{1}-\mathsf{CA}_0$. This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies $\Pi_{2}^{1}$ comprehension.

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Carl Mummert. Stephen G. Simpson. "Reverse mathematics and $\Pi_{2}^{1}$ comprehension." Bull. Symbolic Logic 11 (4) 526 - 533, December 2005. https://doi.org/10.2178/bsl/1130335208

Information

Published: December 2005
First available in Project Euclid: 26 October 2005

zbMATH: 1106.03050
MathSciNet: MR2198712
Digital Object Identifier: 10.2178/bsl/1130335208

Rights: Copyright © 2005 Association for Symbolic Logic

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Vol.11 • No. 4 • December 2005
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