Series: Penn State Logic Seminar Date: Tuesday, April 12, 2005 Time: 2:30 - 3:45 PM Place: 103 Pond Laboratory Speaker: Carl Mummert, Penn State, Mathematics Title: The Reverse Mathematics of Urysohn's Theorem, part 1 Abstract: Urysohn's Theorem states that a regular, second-countable topological space is metrizable. Because this theorem is part of the basic knowledge of most mathematicians, it is natural to analyze Urysohn's theorem from the point of view of Reverse Mathematics, a program in mathematical logic whose goal is to determine which set existence axioms are required to prove theorems of core mathematics. To analyze Urysohn's theorem, we formalize it in second-order arithmetic. We represent topological spaces as spaces of maximal filters on partially ordered sets. If P is a poset, we let MF(P) denote the set of maximal filters on P and topologize MF(P) with the basis {N_p : p in P} where N_p = { f in MF(P) | p in f }. Spaces of the form MF(P) are called MF spaces; if P is countable then MF(P) is called countably based. The class of countably based MF spaces includes all the Polish spaces and many nonmetrizable spaces. We present a formalization of countably based MF spaces in second-order arithmetic. We use this formalization to show that Urysohn's metrization theorem for countably based MF spaces is equivalent to Pi^1_2 - CA_0 over Pi^1_1 - CA_0. This is the first Reverse Mathematics result which shows that a well-known theorem of core mathematics is equivalent to Pi^1_2 - CA_0.