Baire property

As Baire property (or property of Baire , Eng. Property of Baire or Baire property , according to René-Louis Baire ) refers in general topology , particularly the descriptive set theory a property of certain benign subsets of a topological space . A set is baire when it differs only a meager amount from an open set .

definition

A subset of a topological space has the Baire property if and only if there is an open set such that the symmetric difference is lean. ${\displaystyle M\subseteq X}$ ${\displaystyle X}$ ${\displaystyle O\subseteq X}$ ${\displaystyle M\bigtriangleup O}$

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 boundary of a closed set is nowhere dense and therefore lean, because if it is dense in an open set , it is . Thus, there is no element of having an open environment in . But it is open, so it has to be empty and therefore not sealed anywhere. ${\displaystyle X}$ ${\displaystyle \partial A}$ ${\displaystyle A}$ ${\displaystyle O}$ ${\displaystyle O\setminus \partial A\subset X\setminus A}$ ${\displaystyle O\setminus \partial A}$ ${\displaystyle X\setminus A}$ ${\displaystyle X\setminus A}$ ${\displaystyle O}$ ${\displaystyle \partial A}$

Every Borel set has the Baire property. This follows by (countable) transfinite induction over the Borel hierarchy : If all sets from have the Baire property for all ordinal numbers , then every -set as a countable union of sets with the Baire property also has the Baire property. If every -set has the Baire property, then every -set also has the Baire property, because it is the complement of a -set, and thus the complement of a set that differs only a meager amount from an open set. Therefore it differs from a closed set - the complement of said open set - only by that meager set and thus also has the Baire property. It followed that every Borel set has the Baire property, analogously one can deduce that the sets with the Baire property form a σ-algebra . ${\displaystyle \Pi _{m}^{0}}$ ${\displaystyle m<n}$ ${\displaystyle \Sigma _{n}^{0}}$ ${\displaystyle \Sigma _{n}^{0}}$ ${\displaystyle \Pi _{n}^{0}}$ ${\displaystyle \Sigma _{n}^{0}}$

This does not apply to the projective hierarchy . The existence of projective sets that do not have the Baire property is independent of the system of axioms ZFC . The non-existence of such sets follows, for example, from the axiom of projective determinacy , which follows from the existence of Woodin cardinal numbers . The existence of a projective set ( ) without the Baire property, on the other hand, follows, for example, from the constructability axiom that goes back to Kurt Gödel . Analytical and co-analytical quantities, on the other hand, have the Baire property in ZFC, while this can no longer be shown for quantities. ${\displaystyle \Delta _{2}^{1}}$ ${\displaystyle \Delta _{2}^{1}}$

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

Individual evidence

1. ↑ ^{a b c} Descriptive Set Theory (PDF; 643 kB) , lecture notes by David Marker, 2002
2. ↑ W. Hugh Woodin , Strong Axioms of Infinity and the search for V (PDF; 160 kB)
3. ^ Haim Judah and Otmar Spinas: Large cardinals and projective sets
4. ^ Haim Judah, Saharon Shelah, Baire property and Axiom of Choice