# GetWiki

*Baire category theorem*

ARTICLE SUBJECTS

being →

database →

ethics →

fiction →

history →

internet →

language →

linux →

logic →

method →

news →

policy →

purpose →

religion →

science →

software →

truth →

unix →

wiki →

ARTICLE TYPES

essay →

feed →

help →

system →

wiki →

ARTICLE ORIGINS

critical →

forked →

imported →

original →

Baire category theorem

[ temporary import ]

**please note:**

- the content below is remote from Wikipedia

- it has been imported raw for GetWiki

**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 theorem was proved by French mathematician RenÃ©-Louis Baire in his 1899 doctoral thesis.

## Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets {U_n}_{n=1}^infty, their intersection textstylebigcap_{n=1}^infty U_n is dense.- (
**BCT1**) Every complete metric space is a Baire space. More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space. Thus every completely metrizable topological space is a Baire space. - (
**BCT2**) Every locally compact Hausdorff space is a Baire space. The proof is similar to the preceding statement; the finite intersection property takes the role played by completeness.

- (
**BCT3**) A non-empty complete metric space, or any of its subsets with nonempty interior, is not the countable union of nowhere-dense sets.

*non-empty*interior.

## Relation to the axiom of choice

The proofs of**BCT1**and

**BCT2**for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.Blair 1977A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.Levy 1979, p. 212 This restricted form applies in particular to the real line, the Baire space Ï‰Ï‰, the Cantor space 2Ï‰, and a separable Hilbert space such as

*L*2(

*R*n).

## Uses of the theorem

**BCT1**is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.

**BCT1**also shows that every complete metric space with no isolated points is uncountable. (If

*X*is a countable complete metric space with no isolated points, then each singleton {

*x*} in

*X*is nowhere dense, and so

*X*is of first category in itself.) In particular, this proves that the set of all real numbers is uncountable.

**BCT1**shows that each of the following is a Baire space:

- The space mathbb{R} of real numbers
- The irrational numbers, with the metric defined by d(x,y) = tfrac{1}{n+1}, where n is the first index for which the continued fraction expansions of x and y differ (this is a complete metric space)
- The Cantor set

**BCT2**, every finite-dimensional Hausdorff manifold is a Baire space, since it is locally compact and Hausdorff. This is so even for non-paracompact (hence nonmetrizable) manifolds such as the long line.

**BCT**is used to prove Hartog's theorem, a fundamental result in the theory of several complex variables.

## Proof

The following is a standard proof that a complete pseudometric space scriptstyle X is a Baire space.Let U_n be a countable collection of open dense subsets. We want to show that the intersection bigcap U_n is dense. A subset is dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set W in X has a point x in common with all of the U_n. Since U_1 is dense, W intersects U_1; thus, there is a point x_1 and 0 ;**- content above as imported from Wikipedia**

- "

- time: 11:37pm EDT - Sat, Aug 24 2019

- "

__Baire category theorem__" does not exist on GetWiki (yet)- time: 11:37pm EDT - Sat, Aug 24 2019

[ this remote article is provided by Wikipedia ]

LATEST EDITS [ see all ]

GETWIKI 09 JUL 2019

GETWIKI 09 MAY 2016

GETWIKI 18 OCT 2015

GETWIKI 20 AUG 2014

GETWIKI 19 AUG 2014

© 2019 M.R.M. PARROTT | ALL RIGHTS RESERVED