Time:  12  1:20 p.m. 
Room: 
OSC 201

Speaker: 
Stephen Simpson Professor Department of Mathematics Pennsylvania State University 
Title: 
Reverse mathematics and Π^{1}_{2} comprehension 
Abstract: 
This is joint work with Carl Mummert. We initiate the reverse
mathematics of general topology. We show that a certain metrization
theorem is equivalent to Π^{1}_{2} comprehension. An MF space is defined
to be a topological space of the form MF(P) with topology generated by
{ N_{p}  p ε P }. Here P is a poset, MF(P) is the set of maximal
filters on P, and N_{p} = { 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 ACA_{0} of second order arithmetic. One can prove within
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
Π^{1}_{2} CA_{0}. The equivalence is proved in the weaker system
Π^{1}_{1} CA_{0}. This is the first example of a theorem of core
mathematics which is provable in second order arithmetic and implies
Π^{1}_{2} comprehension.

Organizer's note:  Lunch will be provided.
