universally Baire set

In the mathematical field of descriptive set theory, a set of real numbers (or more generally a 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.

Definition

A subset A of the Baire space is universally Baire if it has 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 λ^ω to the Baire space, the preimage of A under f has the property of Baire.

{{cite book|editor1-first=Joan|editor1-last=Bagaria|editor2-first=Stevo|editor2-last=Todorcevic|editor2-link=Stevo Todorčević|title=Set Theory: Centre de Recerca Matemàtica Barcelona, 2003-2004 |series=Trends in Mathematics |isbn=978-3-7643-7691-8 }}
{{cite book |last1=Feng |first1=Qi |first2=Menachem|last2=Magidor|author2-link=Menachem Magidor|first3=Hugh|last3=Woodin|author3-link=Hugh Woodin |editor1-first=H.|editor1-last=Judah|editor2-first= W.|editor2-last=Just|editor3-first= Hugh|editor3-last= Woodin |title=Set Theory of the Continuum |series=Mathematical Sciences Research Institute Publications }}