# Garrett Ervin: Decomposing the real line into two everywhere isomorphic

Mathematical logic seminar – Apr 10 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     Decomposing the real line into two everywhere isomorphic pieces

Abstract:

A dense linear order is said to be homogeneous if it is isomorphic to its restriction to any of its open intervals. The set of rationals ℚ is an example of a homogeneous order, as is the set of irrationals ℝ ∖ ℚ. In general, if X is a homogeneous suborder of the real line ℝ, then ℝ ∖ X is also homogeneous, and there are many examples where both X and ℝ ∖ X are of size continuum. However, it turns out that a homogeneous X can never be isomorphic to ℝ ∖ X. In fact, if ℝ = A ∪ B is any decomposition of ℝ into two disjoint pieces, there is an open interval I such that A restricted to I is not isomorphic to B restricted to I. We will prove this theorem and discuss some related results.