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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
Baire_function
French mathematician (1874–1932)
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
René-Louis_Baire
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
Baire_set
Function studied in real analysis
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
Baire_one_star_function
On topological spaces where the intersection of countably many dense open sets is dense
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
Baire_category_theorem
Function that is continuous everywhere but differentiable nowhere
continuous function proof of existence using Banach's contraction principle. Nowhere monotonic continuous function proof of existence using the Baire category
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
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
List_of_types_of_functions
Concept in topology
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
Baire_space
Indicator function of rational numbers
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
Dirichlet_function
Function that is discontinuous at rationals and continuous at irrationals
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
Thomae's_function
Property of functions which is weaker than continuity
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
Semi-continuity
Concept in set theory
In set theory, the Baire space is the set of all infinite sequences of natural numbers. This space is commonly used in descriptive set theory, to the
Baire_space_(set_theory)
Function in Boolean algebra
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
Parity_function
Measure for Baire sets in mathematics
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
Baire_measure
Analog of Fubini's theorem for arbitrary second countable Baire spaces
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
Kuratowski–Ulam_theorem
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
Universally_Baire_set
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
List_of_incomplete_proofs
List of terms created from a person's name
René Baire, French mathematician – Baire category theorem, Baire function, Baire measure, Baire set, Baire space, Baire space, Property of Baire John
List_of_eponyms_(A–K)
All derivatives have the intermediate value property
necessary condition for a function to be a derivative, but it is not sufficient. Every derivative of a real function is also of Baire class one, and the set
Darboux's_theorem_(analysis)
"Small" subset of a topological space
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
Meagre_set
Hierarchy of complexity classes for formulas defining sets
takes each function from ω {\displaystyle \omega } to ω {\displaystyle \omega } to the characteristic function of its graph. A subset of Baire space is
Arithmetical_hierarchy
Concept in mathematical analysis
differentiable function (and more generally, of any Baire class one function) is a Gδ subset of the real line. By definition, for any Pompeiu function, this set
Pompeiu_derivative
About mathematical functions
(1972). "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue". Archive
History of the function concept
History_of_the_function_concept
Concept in mathematical logic and set theory
\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
Analytical_hierarchy
Index of articles associated with the same name
uses the Baire category theorem. In calculus, part of the inverse function theorem which states that a continuously differentiable function between Euclidean
Open_mapping_theorem
Topological vector spaces
C_{\text{c}}^{k}(U)} is of the first category in itself. It follows from Baire's theorem that C c k ( U ) {\displaystyle C_{\text{c}}^{k}(U)} is not metrizable
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Multivariate functions can be written using univariate functions and summing
theorem (or superposition theorem) states that every multivariate continuous function f : [ 0 , 1 ] n → R {\displaystyle f\colon [0,1]^{n}\to \mathbb {R} } can
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Function spaces generalizing finite-dimensional p norm spaces
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
Lp_space
Statement about linear functionals and measures
compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures
Riesz–Markov–Kakutani representation theorem
Riesz–Markov–Kakutani_representation_theorem
Function from sets to numbers
containing τ {\displaystyle \tau } ). a Baire measure if it is a measure defined on the σ-algebra of all Baire sets. locally finite if for every point
Set_function
Normed vector space that is complete
subsets of the Baire class, see Bourgain, Jean; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math
Banach_space
Countable intersection of open sets
{\displaystyle \mathbb {R} } , a violation of the Baire category theorem. The continuity set of any real valued function is a Gδ subset of its domain (see the "Properties"
Gδ_set
Area of mathematics
{\displaystyle A(U)} is open in Y {\displaystyle Y} ). The proof uses the Baire category theorem, and completeness of both X {\displaystyle X} and Y {\displaystyle
Functional_analysis
Conditions for switching order of integration in calculus
distributions, that is, generalized functions Kuratowski–Ulam theorem – analog of Fubini's theorem for arbitrary second countable Baire spaces Symmetry of second
Fubini's_theorem
Function of two vectors linear in each argument
conditions for a separately continuous bilinear map to be continuous. If X is a Baire space and Y is metrizable then every separately continuous bilinear map
Bilinear_map
Descriptive set theory concept
Baire, and the perfect set property. In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire space
Pointclass
Counterintuitive mathematical object
least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically
Pathological_(mathematics)
Locally convex topological vector space that is also a complete metric space
{\displaystyle X} is a Fréchet space if and only if it is both a webbed space and a Baire space. In contrast to Banach spaces, the complete translation-invariant
Fréchet_space
Branch of topology
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
General_topology
Generalization of mass, length, area and volume
include: Borel measure, Jordan measure, ergodic measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure. In physics an example
Measure_(mathematics)
American mathematician (1922–2007)
1966, pp. 475–476. Edgar Lorch and Hing Tong, "Continuity of Baire Functions and Order of Baire Sets", Indiana University Mathematics Journal, 16: 1967, pp
Hing_Tong
Topics referred to by the same term
Amor Con La Ropa One Star Hotel, a Philadelphia-based rock band Baire one star function, in mathematical real analysis Lone Star (disambiguation) OnStar
One_star
Branch of mathematics
measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized
Mathematical_analysis
René Baire was among the first to systematically study the relationship between separate and joint continuity in 1899, for real-valued functions of real
Namioka's_theorem
A=f^{-1}[B]} for some function f {\displaystyle f} in F. Any such class of functions again determines a preorder on the subsets of Baire space. Degrees given
Wadge_hierarchy
Subset whose closure is the whole space
X . {\displaystyle X.} This fact is one of the equivalent forms of the Baire category theorem. The real numbers with the usual topology have the rational
Dense_set
Rules out assigning to arbitrary functions their computational complexity
almost all total computable predicates are speedup-able, in the sense of the Baire category theorem. Gödel's speed-up theorem Bridges, Douglas S. (1994), "Abstract
Blum's_speedup_theorem
Objects that generalize functions
analysis, distributions are a type of generalized function that makes it possible to differentiate functions whose derivatives do not exist in the classical
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Axiom of set theory
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
Axiom_of_choice
Mathematical property of a space
metrizable neighbourhood. Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection
Topological_property
Descriptive set theory concept
say Baire space or Cantor space or the real line. There is a close relationship between the relativized analytical hierarchy on subsets of Baire space
Projective_hierarchy
Topology in the study of subharmonic functions
does at least have a few 'nicer' properties: The fine topology has the Baire property. The fine topology in R n {\displaystyle \mathbb {R} ^{n}} is locally
Fine topology (potential theory)
Fine_topology_(potential_theory)
Property holding for typical examples
of Cr mappings between M and N, is a Baire space, hence any residual set is dense. This property of the function space is what makes generic properties
Generic_property
Topology on the real numbers
is generated by a quasimetric. R l {\displaystyle \mathbb {R} _{l}} is a Baire space. R l {\displaystyle \mathbb {R} _{l}} does not have any connected
Lower_limit_topology
Metric geometry
bounded functions f : X → M {\displaystyle f:X\to M} is a closed subspace of B ( X , M ) {\displaystyle B(X,M)} and hence also complete. The Baire category
Complete_metric_space
Left-invariant (or right-invariant) measure on locally compact topological group
authors define a Haar measure on Baire sets rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular
Haar_measure
Mathematical set whose closure has empty interior
meagre set. Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental results
Nowhere_dense_set
Algorithm for linear programming
random matrices. Another approach to studying "typical phenomena" uses Baire category theory from general topology, and to show that (topologically)
Simplex_algorithm
Covering space Atlas Limit point Net Filter Ultrafilter Baire category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set Compact
List of general topology topics
List_of_general_topology_topics
led to adopt ZF + DC + BP (dependent choice is a weakened form and the Baire property is a negation of strong AC) as his axioms to prove the Garnir–Wright
Discontinuous_linear_map
Class of mathematical sets
Hausdorff). Borel hierarchy – Mathematical logic hierarchy Borel isomorphism Baire set Cylindrical σ-algebra Descriptive set theory – Subfield of mathematical
Borel_set
Basic subset of a topological space
{\displaystyle x} (in X {\displaystyle X} ). almost open and is said to have the Baire property if there exists an open subset U ⊆ X {\displaystyle U\subseteq
Open_set
Meagre set Nowhere dense set Bounded set Totally bounded set Borel set Baire set Measurable set, Non-measurable set Universally measurable set Negligible
List_of_types_of_sets
French mathematician (1875–1941)
development of Lebesgue integration, dealt with the extension of Baire's theorem to functions of two variables. The next five dealt with surfaces applicable
Henri_Lebesgue
Position in Gaelic games
In Gaelic games, the goalkeeper (Irish: cúl báire, báireoir) is the player responsible for defending the goal — the area between the goalposts and below
Goalkeeper_(Gaelic_games)
Mathematical problem in classical harmonic analysis
arguments invoking the Baire category theorem, this proof is nonconstructive. It shows that the family of continuous functions whose Fourier series converges
Convergence_of_Fourier_series
Theorem stating that pointwise boundedness implies uniform boundedness
space X {\displaystyle X} enables the following short proof, using the Baire category theorem. Proof Suppose X {\displaystyle X} is a Banach space and
Uniform_boundedness_principle
French mathematician (1871–1956)
Marbo. Émile Borel died in Paris on 3 February 1956. Along with René-Louis Baire and Henri Lebesgue, Émile Borel was among the pioneers of measure theory
Émile_Borel
Linear partial differential equation with no solutions
{\displaystyle \mathbb {R} \times \mathbb {C} } . The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of
Lewy's_example
Estimate of time taken for running an algorithm
Zoo: Class SUBEXP: Deterministic Subexponential-Time Moser, P. (2003). "Baire's Categories on Small Complexity Classes". In Andrzej Lingas; Bengt J. Nilsson
Time_complexity
Possible axiom for set theory
constant. Formally, f is the Minkowski question mark function, {0, 1}ω is the Cantor space and ωω is the Baire space.) Observe the equivalence relation on {0
Axiom_of_determinacy
Real number that can be computed within arbitrary precision
available at the time. Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms
Computable_number
Series of four mathematics textbooks
spaces, distributions, the Baire category theorem, probability theory including Brownian motion, the theory of functions of several complex variables
Princeton Lectures in Analysis
Princeton_Lectures_in_Analysis
maigre) of the Baire category theorem. Igusa zeta-function An Igusa zeta-function, named for Jun-ichi Igusa, is a generating function counting numbers
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Weak form of the axiom of choice
ISBN 0-387-90670-3. "The Baire category theorem implies the principle of dependent choices." Blair, Charles E. (1977). "The Baire category theorem implies
Axiom_of_dependent_choice
Mathematical concept
limited to act on functions on the unit interval of the real number line. More broadly, since the Gauss map is the shift operator on Baire space N ω {\displaystyle
Gauss–Kuzmin–Wirsing_operator
communication workers federation By 2012, NCEW had 26,000 members. Tekeste Baire had been the Secretary-General of the organization since 1994 until his
National Confederation of Eritrean Workers
National_Confederation_of_Eritrean_Workers
Mathematical game on a topological space
natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties
Topological_game
Subfield of set theory
given a particular sequence of plays. More formally, consider a subset A of Baire space; recall that the latter consists of all ω-sequences of natural numbers
Determinacy
the Rothberger property can be characterized using continuous functions into the Baire space N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} . A subset
Rothberger_space
Belgian mathematician (1866–1962)
infinitésimale, Tome II Intégrales de Lebesgue, fonctions d´ensemble, classes de Baire, 2nd edition 1934, Reprint by Jacques Gabay, ISBN 2-87647-159-0 Le potentiel
Charles-Jean de La Vallée Poussin
Charles-Jean_de_La_Vallée_Poussin
Theorem in descriptive set theory
and when A is the set of natural numbers, it is the ordinary topology on Baire space. The set Aω can be viewed as the set of paths through a certain tree
Borel_determinacy_theorem
Set of real numbers that is not Lebesgue measurable
λ ( V ) {\displaystyle \lambda (V)} . No Vitali set has the property of Baire. By modifying the above proof, one shows that each Vitali set has Banach
Vitali_set
Type of topological vector space
characterization of Baire TVSs proved by Saxon [1974], who proved that a TVS Y {\displaystyle Y} with a topology that is not the indiscrete topology is a Baire space
Barrelled_space
Set of vectors used to define coordinates
Hamel basis of X is necessarily uncountable. This is a consequence of the Baire category theorem. The completeness as well as infinite dimension are crucial
Basis_(linear_algebra)
Isomorphism of differentiable manifolds
strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, still Baire. Fixing a Riemannian metric on M {\displaystyle
Diffeomorphism
Generalized notion of measure in mathematics
then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued functions on X, by the Riesz–Markov–Kakutani
Signed_measure
1985 book by Pauline Kael
The Return of the Soldier, A Private Function, The Purple Rose of Cairo, Heartbreakers, Lost in America, Ghare Baire, Prizzi's Honor, The Shooting Party
State_of_the_Art_(book)
Area of functional analysis and convex analysis
similar representation with a probability measure that vanishes on the Baire subsets of C which contain no extreme points. In addition to the existence
Choquet_theory
Subject in mathematics
{\displaystyle X'} . the Baire σ-algebra B 0 ( X ) {\displaystyle {\mathcal {B}}_{0}(X)} : is generated by all continuous functions C ( X , R ) {\displaystyle
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Set of points on a line segment with certain topological properties
itself, since it is a Baire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not
Cantor_set
Mathematical system
include coanalytic perfect subset property, measurability and the property of Baire for Σ 2 1 {\displaystyle \Sigma _{2}^{1}} sets, Π 3 1 {\displaystyle \Pi
Second-order_arithmetic
Generalization of Turing computability
indices of recursive ordinals. The set of elements of Baire space that are the characteristic functions of a well ordering of the natural numbers (using an
Hyperarithmetical_theory
Generalization of Rice's theorem
(U(k)\in P)\neq B} . The set of total computable functions can be viewed as a subspace of the Baire space T ⊆ N N {\displaystyle {\mathcal {T}}\subseteq
Rice–Shapiro_theorem
Type of topological space in mathematics
particular every locally compact Hausdorff space, is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every countable
Locally_compact_space
Mathematical function revertible near each point
between two Hausdorff second-countable spaces where X {\displaystyle X} is a Baire space and Y {\displaystyle Y} is a normal space. If every fiber of f {\displaystyle
Local_homeomorphism
Baire 1. René-Louis Baire 2. A subset of a topological space has the Baire property if it differs from an open set by a meager set 3. The Baire space
Glossary_of_set_theory
Type of convergence
first introduced by René Baire in 1908 in his book Leçons sur les théories générales de l'analyse. Given a set S and functions f n : S → C {\displaystyle
Normal_convergence
Branch of mathematical logic
single point in its intersection; the real numbers are not countable). The Baire category theorem for a complete separable metric space (the separability
Reverse_mathematics
All points in the topological closure not belonging to the interior
for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces. A set is the boundary of some open set if and only if it is closed
Boundary_(topology)
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BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
BAIRE FUNCTION
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