Baire category theorem

The Baire category theorem (BCT) is an important result in

.

Versions of the Baire category theorem were first proved independently in 1897 by

real line

${\displaystyle \mathbb {R} }$ and in 1899 by Baire^[1] for Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.^[2] The more general statement for completely metrizable spaces was first shown by Hausdorff^[3] in 1914.

Statement

A Baire space is a topological space ${\displaystyle X}$ in which every

is dense in ${\displaystyle X.}$ See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application.

• (BCT1) Every
  completely metrizable topological space is a Baire space.^[7]
• (BCT2) Every
  locally compact Hausdorff space is a Baire space.^[9][7]

Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the

). See Steen and Seebach in the references below.

Relation to the axiom of choice

The proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice, a weak form of the axiom of choice.^[10]

A 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.^[11] This restricted form applies in particular to the

${\displaystyle \omega ^{\omega },}$ the Cantor space ${\displaystyle 2^{\omega },}$ and a separable Hilbert space such as the ${\displaystyle L^{p}}$-space ${\displaystyle L^{2}(\mathbb {R} ^{n})}$.

Uses

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 nonempty complete metric space with no

. (If ${\displaystyle X}$ is a nonempty countable metric space with no isolated point, then each singleton ${\displaystyle \{x\}}$ in ${\displaystyle X}$ is , and ${\displaystyle X}$ is meagre 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 ${\displaystyle \mathbb {R} }$ of real numbers
• The irrational numbers, with the metric defined by ${\displaystyle d(x,y)={\tfrac {1}{n+1}},}$ where ${\displaystyle n}$ is the first index for which the continued fraction expansions of ${\displaystyle x}$ and ${\displaystyle y}$ differ (this is a complete metric space)
• The Cantor set

By BCT2, every finite-dimensional Hausdorff

.

BCT is used to prove

Hartogs's theorem

, a fundamental result in the theory of several complex variables.

BCT1 is used to prove that a Banach space cannot have countably infinite dimension.

Proof

(BCT1) The following is a standard proof that a complete pseudometric space ${\displaystyle X}$ is a Baire space.^[6]

Let ${\displaystyle U_{1},U_{2},\ldots }$ be a countable collection of open dense subsets. It remains to show that the intersection ${\displaystyle U_{1}\cap U_{2}\cap \ldots }$ 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 suffices to show that any nonempty open subset ${\displaystyle W}$ of ${\displaystyle X}$ has some point ${\displaystyle x}$ in common with all of the ${\displaystyle U_{n}}$. Because ${\displaystyle U_{1}}$ is dense, ${\displaystyle W}$ intersects ${\displaystyle U_{1};}$ consequently, there exists a point ${\displaystyle x_{1}}$ and a number ${\displaystyle 0<r_{1}<1}$ such that:

where ${\displaystyle B(x,r)}$ and ${\displaystyle {\overline {B}}(x,r)}$ denote an open and closed ball, respectively, centered at ${\displaystyle x}$ with radius ${\displaystyle r.}$ Since each ${\displaystyle U_{n}}$ is dense, this construction can be continued recursively to find a pair of sequences ${\displaystyle x_{n}}$ and ${\displaystyle 0<r_{n}<{\tfrac {1}{n}}}$ such that:

(This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at ${\displaystyle x_{n}}$.) The sequence ${\displaystyle \left(x_{n}\right)}$ is Cauchy because ${\displaystyle x_{n}\in B\left(x_{m},r_{m}\right)}$ whenever ${\displaystyle n>m,}$ and hence ${\displaystyle \left(x_{n}\right)}$ converges to some limit ${\displaystyle x}$ by completeness. If ${\displaystyle n}$ is a positive integer then ${\displaystyle x\in {\overline {B}}\left(x_{n},r_{n}\right)}$ (because this set is closed). Thus ${\displaystyle x\in W}$ and ${\displaystyle x\in U_{n}}$ for all ${\displaystyle n.}$ ${\displaystyle \blacksquare }$

There is an alternative proof using

(BCT2) The proof that a

locally compact regular

space ${\displaystyle X}$ is a Baire space is similar.

locally compact Hausdorff

spaces is a special case, as such spaces are regular.

Notes

References

• OCLC 246032063
  .
• ISBN 3-88538-006-4
  .
• OCLC 338047
  .
• ISBN 0-486-42079-5
  .
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• OCLC 175294365
  .
• ISBN 0-486-68735-X
  (Dover edition).
• Willard, Stephen (2004) [1970]. General Topology.
  OCLC 115240
  .

External links

• Encyclopaedia of Mathematics article on Baire theorem
• Tao, T. (1 February 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences".