Property of Baire

As Baire property (or property of Baire, Eng. Property of Baire or Baire property, by René Louis Baire ) is referred to in the general topology and in particular the descriptive set theory, a property of certain benign subsets of a topological space. A lot has the Baire property if it differs by only a meager amount of an open set.

Definition

A subset of a topological space if and only has the Baire property if there exists an open set such that the symmetric difference is lean.

Relation to the projective hierarchy and the Borel hierarchy

Every closed set in a topological space has the property of Baire, this can be shown as follows: The edge of a closed set is nowhere dense and thin thus, because he is close, is in an open set so. Thus no element of is with an open environment. But is open, thus must be empty and thus nowhere dense.

Every Borel set has the Baire property. This follows by ( countable ) transfinite induction on the Borel hierarchy: Do all quantities from the Baire property for all ordinals, as well as any amount than countable union of sets with the Baire property, the Baire property. Has any amount the Baire property, as well as any amount has the Baire property, because it is the complement of a quantity, and thus complement an amount that differs by only a meager amount of an open set. Therefore, it differs from a closed set - the complement of said open quantity - only to those very meager amount and thus also has the Baire property. It followed that any Borel set has the Baire property, analogously, one can deduce that the sets form with the Baire property is a σ - algebra.

This does not apply the projective hierarchy. The existence of projective amounts that do not have the Baire property is independent of the axiom system ZFC. The non- existence of such quantities as follows from the axiom of projective determinacy, which follows from the existence of a Woodin cardinal number. The existence of a projective set ( ) without the Baire property follows from the contrast about going back to Kurt Godel Konstruierbarkeitsaxiom. Analytical and koanalytische quantities have, however, in ZFC, the Baire property, while it no longer can be shown for quantities.

The existence of a set without the Baire property follows from the axiom of choice, but not from ZF without the axiom of choice.