functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. Baire functions of...

les fonctions de variables réelles ("On the Functions of Real Variables") in 1899. The son of a tailor, Baire was one of three children from a poor working-class...

compactly supported continuous function on such a space is integrable with respect to any finite Baire measure. Every Baire set is a Borel set. The converse...

shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous...

Measurable function: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function called...

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...

The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient...

A Baire one star function is a type of function studied in real analysis. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is in class...

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. Assume throughout that X...

set. This would contradict the Baire category theorem: because the reals form a complete metric space, they form a Baire space, which cannot be meager...

continuous function proof of existence using Banach's contraction principle. Nowhere monotonic continuous function proof of existence using the Baire category...

In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space...

the property of Baire and thus that no infinite parity function exists; this holds in the Solovay model, for instance. Walsh function, a continuous equivalent...

In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In...

René Baire, French mathematician – Baire category theorem, Baire function, Baire measure, Baire set, Baire space, Baire space, Property of Baire John...

Fubini's theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and let...

Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of ℓ ∞ {\displaystyle \ell ^{\infty }} is ℓ...

(1972). "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue". Archive...

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...

takes each function from ω {\displaystyle \omega } to ω {\displaystyle \omega } to the characteristic function of its graph. A subset of Baire space is...

sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental...

\Delta _{n}^{1}} . A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from ω {\displaystyle \omega...

Lebesgue tried to prove the (correct) result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection...

_{i}(y)){\Big |}\in [0,6/7]} Iterating the above construction, then applying the Baire category theorem, we find that the following kind of 5-tuples are open and...

uses the Baire category theorem. In calculus, part of the inverse function theorem which states that a continuously differentiable function between Euclidean...

least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically...

regulated function has a well-defined Riemann integral. Remark: By the Baire Category theorem the set of points of discontinuity of such function F σ {\displaystyle...

countably many nowhere dense sets is empty. Any open subspace of a Baire space is itself a Baire space. A continuum (pl continua) is a nonempty compact connected...

real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of...

subsets of the Baire class, see Bourgain, Jean; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math...