The Reverse Mathematics of Urysohn’s Theorem, part 1

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.