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Baire Category Theorem


Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.

Precisely stated, the theorem says that if a space S is either a complete metric space or a locally compact T2-space, then the intersection of every countable collection of dense open subsets of S is necessarily dense in S.

The above-mentioned interplay with first and second category sets can be summarized by a single corollary, namely that spaces S that are either complete metric spaces or locally compact Hausdorff spaces are of second category in themselves. To see that this follows from the above-stated theorem, let S be either a complete metric space or a locally compact Hausdorff space and note that if {E_i}={E_i}_(i in N) is a countable collection of nowhere dense subsets of S and if V_i denotes the complement in S of the closure E^__i of E_i, then each set V_i is necessarily dense in S. Because of the theorem, it follows that the intersection of all the sets V_i must be nonempty (and indeed must be dense in S), thereby proving that S cannot be written as the union of the sets E_i. In particular, such spaces S cannot be written as the countable union of sets which are nowhere dense in themselves and are therefore second category sets relative to themselves.


See also

Complete Metric Space, Countable Set, First Category, Meager Set, Metric Space, Nonmeager Set, Nowhere Dense, Second Category, T2-Space

This entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Referenced on Wolfram|Alpha

Baire Category Theorem

Cite this as:

Stover, Christopher. "Baire Category Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BaireCategoryTheorem.html

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