Baire space

In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.

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Motivation

In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets, smooth curves in the plane, and proper affine subspaces in a Euclidean space. A topological space is a Baire space if it is "large", meaning that it is not a countable union of negligible subsets. For example, the three dimensional Euclidean space is not a countable union of its affine planes.

Definition

The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition which is closer to the definition originally given by Baire.

Modern definition

A topological space is called a Baire space if the union of any countable collection of closed sets with empty interior has empty interior.

This definition is equivalent to each of the following conditions:

Historical definition

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows.

A subset of a topological space X is called

The definition for a Baire space can then be stated as follows: a topological space X is a Baire space if every non-empty open set is of second category in X. This definition is equivalent to the modern definition.

A subset A of X is comeagre if its complement X\setminus A is meagre. A topological space X is a Baire space if and only if every comeager subset of X is dense.

Examples

\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} \left(r_{n}-{1 \over 2^{n%2Bm} }, r_{n}%2B{1 \over 2^{n%2Bm}}\right)
where  \left\{r_{n}\right\}_{n=1}^{\infty} is a sequence that enumerates the rational numbers.

Baire category theorem

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis.

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

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. For example, the long line is of second category.

Properties

See also

References