Baire space

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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]

Motivation[edit]

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.

Definition[edit]

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:

  1. Every non-empty open subset of is a nonmeager subset of ;[5]
  2. Every comeagre subset of is dense in ;
  3. The union of any countable collection of closed nowhere dense subsets (i.e. each closed subset has empty interior) has empty interior;[5]
  4. Every intersection of countably many dense open sets in is dense in ;[5]
  5. 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;
  6. 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."

Sufficient conditions[edit]

Baire category theorem[edit]

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 shows that each of the following is a Baire 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.

Other sufficient conditions[edit]

  • 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]

Examples[edit]

  • 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 :
    where is a sequence that enumerates the rational numbers.
  • 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.

Non-example[edit]

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.

Properties[edit]

  • 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.

See also[edit]

Citations[edit]

  1. ^ Munkres 2000, p. 295.
  2. ^ Köthe 1979, p. 25.
  3. ^ Munkres 2000, p. 296.
  4. ^ Haworth & McCoy 1977, p. 5.
  5. ^ a b c d e f g Narici & Beckenstein 2011, pp. 371–423.
  6. ^ Wilansky 2013, p. 60.

References[edit]

  • Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1–123.
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • Munkres, James R. (2000). 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) [1969]. 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

External links[edit]