On the failure of the first principle of separation for coanalytic sets

Abstract

In this note we present a new example of a pair of disjoint coanalytic sets which are not Borel separable, i.e., coanalytic sets D D and H H such that D ∩ H = ϕ D \cap H = \phi and such that there is no Borel set E E for which D ⊆ E D \subseteq E and E ∩ H = ϕ E \cap H = \phi .