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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.
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.
Because is a Baire space, the set := = / = = / is a dense subset of (which means that like its subset , cannot possibly be nowhere dense in ) with Lebesgue measure that is also a nonmeager subset of (that is, is of the second category in ), which makes a comeager subset of whose interior in is also empty; however, is nowhere dense in if and ...
Following Uzawa's theorem, many mathematical economists consider proving existence a deeper result than proving the two Fundamental Theorems. Another method of proof of existence, global analysis, uses Sard's lemma and the Baire category theorem; this method was pioneered by Gérard Debreu and Stephen Smale.
The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem. On every infinite-dimensional topological vector space there is a discontinuous linear map .
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ 1 and 2 ℵ 0 .