Reverse mathematics and $\Pi_{2}^{1}$ comprehension

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.