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 λω to the Baire space, the preimage of A under f has the property of Baire.
Read more about this topic: Universally Baire Set
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