In the mathematical field of
descriptive set theory, a set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (or more generally a subset of the
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
or
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
) is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in
Ω-logic
In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H_. Just as the axiom of projective determinacy yields a canoni ...
, a very strong logical system invented by
W. Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
and the centerpiece of his argument against the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
of
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
.
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
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
''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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
Ω, and every
continuous function ''f'' from Ω to the Baire space, the
preimage of ''A'' under ''f'' has the
property of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such th ...
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
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Descriptive set theory
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