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The Baire category theorem (BCT) is an important result in general topology and functional analysis.The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense).
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. (BCT1) Every complete pseudometric space is a Baire space.[9] [10] In particular, every completely metrizable topological space is a Baire space.
The Banach category theorem [12] states that in any space , the union of any family of open sets of the first category is of the first category. All subsets and all countable unions of meagre sets are meagre. Thus the meagre subsets of a fixed space form a σ-ideal of subsets, a suitable notion of negligible set.
In particular, a Vitali set does not have the property of Baire. [4] Already weaker versions of choice are sufficient: the Boolean prime ideal theorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property. [5]
Proof. Suppose is a Banach space and that for every , ‖ ‖ <.. For every integer , let = { : ‖ ‖}.. Each set is a closed set and by the assumption, =.. By the Baire category theorem for the non-empty complete metric space, there exists some such that has non-empty interior; that is, there exist and > such that ¯ := {: ‖ ‖} .
René-Louis Baire (French:; 21 January 1874 – 5 July 1932) was a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variables réelles ("On the Functions of Real Variables") in 1899.
The Baire category theorem says that every complete metric space is a Baire space. That is, the union of countably many nowhere dense subsets of the space has empty interior. The Banach fixed-point theorem states that a contraction mapping on a complete metric space admits a fixed point.
The Baire Characterisation Theorem states that a real valued function f defined on a Banach space X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K.