Reverse Mathematics and Pi^1_2 Comprehension Stephen G. Simpson Department of Mathematics Pennsylvania State University http://www.personal.psu.edu/t20/ January 17, 2007 Note: This is an abstract of an invited 50-minute talk, to be given January 21, 2007 at the Massachusetts Institute of Technology, as part of a conference in honor of Richard Shore's 60th birthday. 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. If P is a poset, let MF(P) be the space of maximal filters on P. Here MF(P) has the obvious topology generated by basic open sets N_p = { F in MF(P) | p in F }, p in P. An MF space is defined to be a topological space of the form MF(P). If P is countable, we say that MF(P) is 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 regular and is homeomorphic to a countably based MF space. We show that the converse statement, "every regular, countably based MF space 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.