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]

MotivationEdit

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.

DefinitionEdit

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 intersection of countably many dense open sets in   is dense in  ;[5]
  2. Every union of countably many closed subsets of   with empty interior has empty interior;
  3. Every non-empty open subset of   is a nonmeager subset of  ;[5]
  4. Every comeagre subset of   is dense in  ;
  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 conditionsEdit

Baire category theoremEdit

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 conditionsEdit

  • 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 balanced absorbing subset has non-empty interior.[6]

ExamplesEdit

  • 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  .
  • Another large class of Baire spaces are algebraic varieties with the Zariski topology. For example, the space   of complex numbers whose open sets are complements of the vanishing sets of polynomials   is an algebraic variety with the Zariski topology. Usually this is denoted  .
  • 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-exampleEdit

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.

PropertiesEdit

  • 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 alsoEdit

CitationsEdit

  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 Narici & Beckenstein 2011, pp. 371–423.
  6. ^ Wilansky 2013, p. 60.

ReferencesEdit

  • 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. Vol. 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. Vol. 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. Vol. 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. Vol. 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. Vol. 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 linksEdit