# Baire theorem

Baire's theorem on complete spaces: Any countable system of open and everywhere-dense sets in a given complete metric space has a non-empty, and even an everywhere-dense, intersection in this space. An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable sum of its nowhere-dense subsets. Stated by R. Baire [1].

#### References

[1] | R. Baire, Ann. Mat. Pura Appl. , 3 (1899) pp. 67 |

#### Comments

This theorem is also known as the Baire category theorem (cf. [a1], p. 200).

#### References

[a1] | J.L. Kelley, "General topology" , v. Nostrand (1955) |

Baire's theorem on semi-continuous functions: Let be a subset of a metric space , and let . The condition: For any number the set (or, respectively, ) is closed in , is necessary and sufficient for to be semi-continuous from above (or, respectively, from below) on . Demonstrated by R. Baire for [1]. It follows from this theorem that semi-continuous functions belong to the first Baire class (cf. Baire classes). A stronger theorem is valid: A function that is semi-continuous from above (from below) and that does not assume the value () is the limit of a monotone non-increasing (non-decreasing) sequence of continuous functions.

#### References

[1] | R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) |

[2] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) |

*I.A. Vinogradova*

#### Comments

A function in the first Baire class is also called a Baire function.

#### References

[a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1964) |

**How to Cite This Entry:**

Baire theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Baire_theorem&oldid=13173