Baire space

From Academic Kids

For the set theory concept, see Baire space (set theory).

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 Ren-Louis Baire who introduced the concept.



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 the lines.


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.

This definition is equivalent to each of the following conditions:

  • Every intersection of countable dense open sets is dense.
  • The interior of every union of countably many 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.

Historical definition

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows

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 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 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.
<math>\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} (r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}})<math>
where <math> \left\{r_{n}\right\}_{n=1}^{\infty} <math> 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.

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:


  • 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:XY with limit f:XY. 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.

See also


  • Munkres, James, Topology, 2nd edition, Prentice Hall, 2000.
  • Baire, Ren-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, de Baire

fr:espace de Baire ru:Категория Бэра


Academic Kids Menu

  • Art and Cultures
    • Art (
    • Architecture (
    • Cultures (
    • Music (
    • Musical Instruments (
  • Biographies (
  • Clipart (
  • Geography (
    • Countries of the World (
    • Maps (
    • Flags (
    • Continents (
  • History (
    • Ancient Civilizations (
    • Industrial Revolution (
    • Middle Ages (
    • Prehistory (
    • Renaissance (
    • Timelines (
    • United States (
    • Wars (
    • World History (
  • Human Body (
  • Mathematics (
  • Reference (
  • Science (
    • Animals (
    • Aviation (
    • Dinosaurs (
    • Earth (
    • Inventions (
    • Physical Science (
    • Plants (
    • Scientists (
  • Social Studies (
    • Anthropology (
    • Economics (
    • Government (
    • Religion (
    • Holidays (
  • Space and Astronomy
    • Solar System (
    • Planets (
  • Sports (
  • Timelines (
  • Weather (
  • US States (


  • Home Page (
  • Contact Us (

  • Clip Art (
Personal tools