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In mathematics, a topological space is said to be a **Baire space**, if for any given countable collection of closed sets with empty interior in , their union also has empty interior in .^{ [1] } Equivalently, a locally convex space which is not meagre in itself is called a Baire space.^{ [2] } According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space.^{ [3] } Bourbaki coined the term "Baire space".^{ [4] }

In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets in smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is "large", meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space is not a countable union of its affine planes.

The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. A topological space is called a **Baire space** if it satisfies any of the following equivalent conditions:

- Every non-empty open subset of is a nonmeager subset of ;
^{ [5] } - Every comeagre subset of is dense in ;
- The union of any countable collection of closed nowhere dense subsets (i.e. each closed subset has empty interior) has empty interior;
^{ [5] } - Every intersection of countably many dense open sets in is dense in ;
^{ [5] } - The interior (taken in ) of every union of countably many closed nowhere dense sets is empty;
- Whenever the union of countably many closed subsets of has an interior point, then at least one of the closed subsets must have an interior point;
- The complement in of every meagre subset of is dense in ;
^{ [5] } - Every point in has a neighborhood that is a Baire space (according to any defining condition other than this one).
^{ [5] }- So is a Baire space if and only if it is "locally a Baire space."

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.

- (
**BCT1**) Every complete pseudometric space is a Baire space.^{ [5] }More generally, every topological space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, every completely metrizable space is a Baire space. - (
**BCT2**) Every locally compact Hausdorff space (or more generally every locally compact sober space) is a Baire space.

**BCT1** shows that each of the following is a Baire space:

- The space of real numbers
- The space of irrational numbers, which is homeomorphic to the Baire space of set theory
- Every compact Hausdorff space is a Baire space.
- In particular, the Cantor set is a Baire space.

- Indeed, every Polish space.

**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.

- A product of complete metric spaces is a Baire space.
^{ [5] } - A topological vector space is nonmeagre if and only if it is a Baire space,
^{ [5] }which happens if and only if every closed absorbing subset has non-empty interior.^{ [6] }

- The space 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 .
- The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval with the usual topology.
- Here is an example of a set of second category in with Lebesgue measure :
- Note that the space of rational numbers with the usual topology inherited from the real numbers is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.

One of the first non-examples comes from the induced topology of the rationals inside of the real line with the standard euclidean topology. Given an indexing of the rationals by the natural numbers so a bijection and let where which is an open, dense subset in Then, because the intersection of every open set in is empty, the space cannot be a Baire space.

- Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of 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
- Every open subspace of a Baire space is a Baire space.
- Given a family of continuous functions = with pointwise limit If is a Baire space then the points where is not continuous is
*a meagre set*in and the set of points where is continuous is dense in A special case of this is the uniform boundedness principle. - A closed subset of a Baire space is not necessarily Baire.
- The product of two Baire spaces is not necessarily Baire. However, there exist sufficient conditions that will guarantee that a product of arbitrarily many Baire spaces is again Baire.

- ↑ Munkres 2000, p. 295.
- ↑ Köthe 1979, p. 25.
- ↑ Munkres 2000, p. 296.
- ↑ Haworth & McCoy 1977, p. 5.
- 1 2 3 4 5 6 7 8 Narici & Beckenstein 2011, pp. 371-423.
- ↑ Wilansky 2013, p. 60.

In mathematical analysis, a metric space M is called **complete** if every Cauchy sequence of points in M has a limit that is also in M.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In geometry, topology, and related branches of mathematics, a **closed set** is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

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. More precisely, it is a topological space in which every point has a compact neighborhood.

In mathematics, a subset of a topological space is called **nowhere dense** or **rare** if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not.

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 conditions for a topological space to be a Baire space.

In mathematics, specifically in topology, the **interior** of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an **interior point** of S.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In the mathematical fields of general topology and descriptive set theory, a **meagre set** is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. A topological space T is called **meagre** if it is a meager subset of itself; otherwise, it is called **nonmeagre**.

In the mathematical field of topology, a **G _{δ} set** is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with

In mathematics, the **lower limit topology** or **right half-open interval topology** is a topology defined on the set of real numbers; it is different from the standard topology on and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [*a*,*b*), where *a* and *b* are real numbers.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

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 topology induced from that of *X* called the **subspace topology**.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In mathematics, **Baire functions** are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

- Baire, René-Louis (1899), Sur les fonctions de variables réelles,
*Annali di Mat. Ser. 3***3**, 1–123. - Grothendieck, Alexander (1973).
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*Topology*. Prentice-Hall. ISBN 0-13-181629-2. - Khaleelulla, S. M. (1982).
*Counterexamples in Topological Vector Spaces*. Lecture Notes in Mathematics.**936**. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. - Köthe, Gottfried (1983).
*Topological Vector Spaces I*. Grundlehren der mathematischen Wissenschaften.**159**. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. - Köthe, Gottfried (1979).
*Topological Vector Spaces II*. Grundlehren der mathematischen Wissenschaften.**237**. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972. - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics.**8**(Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. - Wilansky, Albert (2013).
*Modern Methods in Topological Vector Spaces*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. - Haworth, R. C.; McCoy, R. A. (1977),
*Baire Spaces*, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk

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