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In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honour of René-Louis Baire who introduced the concept. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
René-Louis Baire (born January 21, 1874, died July 5, 1932) was a French mathematician. ...
Motivation
In a topological space we can think of closed sets with empty interior as points in the space. Ignoring spaces with isolated points, which are their own interior, a Baire space is "large" in the sense that it cannot be constructed as a countable union of its points. A concrete example is a 2-dimensional plane with a countable collection of lines. No matter what lines we choose we cannot cover the space completely with a countable set of them. In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
The empty set is the set containing no elements. ...
In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an...
In mathematics the term countable set is used to describe the size of a set, e. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
Definition The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition which is closer to the definition originally given by Baire.
Modern definition A topological space is called a Baire space if the countable union of any collection of closed sets with empty interior has empty interior. In mathematics the term countable set is used to describe the size of a set, e. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
The empty set is the set containing no elements. ...
In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
This definition is equivalent to each of the following conditions:
- Every intersection of countably many dense open sets is dense.
- The interior of every union of countably many closed nowhere dense sets is empty.
- Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ...
Historical definition In his original definition, Baire defined a notion of category (unrelated to category theory) as follows In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
A subset of a topological space X is called
- nowhere dense in X if the interior of its closure is empty
- of first category or meagre in X if it is a union of countably many nowhere dense subsets
- of second category or nonmeagre in X if it is not of first category in X
The definition for a Baire space can then be stated as follows: a topological space X is a Baire space if every non-empty open set is of second category in X. This definition is equivalent to the modern definition. In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ...
In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...
The empty set is the set containing no elements. ...
In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ...
A subset A of X is comeagre if its complement  is meagre. In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
Examples - The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself. The rational numbers are of first category and the irrational numbers are of second category in R.
- The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval [0, 1] with the usual topology.
- Here is an example of a set of second category in R with Lebesgue measure 0.
-
 - where
 is a sequence that counts the rational numbers. - Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...
In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a singleton is a set with exactly one element. ...
Baire category theorem The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis. The Baire category theorem is an important tool in general topology and functional analysis. ...
In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...
A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
BCT1 shows that each of the following is a Baire space: In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
This word should not be confused with homomorphism. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. For example, the long line is of second category. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...
In mathematics, a Polish space is a separable completely metrisable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...
A metrizable space is a topological space that is homeomorphic to a metric space. ...
In topology, the long line is a topological space analogous to the real line, but much longer. ...
Properties - Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0, 1].
- Given a family of continuous functions fn:X→Y with limit f:X→Y. If X is a Baire space then the points where f is not continuous is meagre in X and the set of points where f is continuous is dense in X.
In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. ...
In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...
In mathematics, an index set is another name for a function domain. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
See also In mathematics, in particular in general topology and set theory, a Banach-Mazur game is a game played between two players, trying to pin down elements in a set (space). ...
In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ...
References - Munkres, James, Topology, 2nd edition, Prentice Hall, 2000.
- Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1--123.
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