Reverse Mathematics and Π 12 Comprehension

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 2 comprehension. An MF space is defined to be a topological space of the form MF(P ) with topology generated by {Np | p ∈ P}. Here P is a poset, MF(P ) is the set of maximal filters on P , and Np = {F ∈ MF(P ) | p ∈ 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 Π 2 -CA0. The equivalence is proved in the weaker system Π 1 -CA0. This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π 2 comprehension.