In the mathematical field of descriptive set theory
, a set of reals (or subset of the Baire space
or Cantor space
) is called universally Baire
if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic
, a very strong logical system invented by W. Hugh Woodin
and the centerpiece of his argument against the continuum hypothesis
of Georg Cantor
A subset A
of the Baire space is universally Baire if it has one of the following equivalent properties:
- For every notion of forcing, there are trees T and U such that A is the projection of the set of all branches through T, and it is forced that the projections of the branches through T and the branches through U are complements of each other.
- For every compact Hausdorff space Ω, and every continuous function f from Ω to the Baire space, the preimage of A under f has the property of Baire in Ω.
- For every cardinal λ and every continuous function f from λω to the Baire space, the preimage of A under f has the property of Baire.
- Set Theory: Centre de Recerca Matemàtica Barcelona, 2003-2004 (Trends in Mathematics).
- Feng, Qi; Menachem Magidor and Hugh Woodin Set Theory of the Continuum (Mathematical Sciences Research Institute Publications).