Title: Reverse Mathematics and Pi^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 Pi^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 in P }. Here P is a poset, MF(P) is the set of maximal
filters on P, and N_p = { F in 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 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
Pi^1_2-CA_0. The equivalence is proved in the weaker system
Pi^1_1-CA_0. This is the first example of a theorem of core
mathematics which is provable in second order arithmetic and implies
Pi^1_2 comprehension.