# Baire theorem

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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

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)