Universally Baire set

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.

Definition

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 λ^ω to the Baire space, the preimage of "A" under "f" has the property of Baire.

References

*cite book |last= |first= |authorlink= |coauthors= |editor=Joan Bagaria and Stevo Todorcevic (eds) |others= |title=Set Theory: Centre de Recerca Matemàtica Barcelona, 2003-2004 (Trends in Mathematics) |origdate= |origyear= |origmonth= |url= |format= |accessdate= |accessyear= |accessmonth= |edition= |series= |volume= |date= |year= |month= |publisher= |location= |language= |isbn=978-3764376918 |oclc= |doi= |id= |pages= |chapter= |chapterurl= |quote= |ref=

*cite book |last=Feng |first=Qi |authorlink= |coauthors=Menachem Magidor and Hugh Woodin |editor=H. Judah, W. Just, and Hugh Woodin |others= |title=Set Theory of the Continuum (Mathematical Sciences Research Institute Publications) |origdate= |origyear= |origmonth= |url= |format= |accessdate= |accessyear= |accessmonth= |edition= |series= |volume= |date= |year= |month= |publisher= |location= |language= |isbn= |oclc= |doi= |id= |pages= |chapter= |chapterurl= |quote= |ref=