In the mathematical field of
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 ...
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 canonic ...
, a very strong logical system invented by
W. Hugh Woodin and the centerpiece of his argument against the
continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
Or equivalently:
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
of
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
.
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, e.g., including only woody plants with secondary growth, only p ...
''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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
Ω, and every
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
''f'' from Ω to the Baire space, the
preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
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 tha ...
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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