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]

Contents

Motivation

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

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. The interior (taken in ) of every union of countably many closed nowhere dense sets is empty;
  6. 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;
  7. The complement in of every meagre subset of is dense in ; [5]
  8. 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

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.

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

Examples

Non-example

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

See also

Citations

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

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References