Baire category theorem

The set of Baire, also called Baire category set or a set of categories, in the topology treats a room property, which belongs to a variety of topological spaces, in particular, the real numbers and Euclidean spaces. This property is in various adjoining areas of mathematics, such as the descriptive set theory, measure theory and functional analysis of considerable importance. A topological space which satisfies the conditions of the theorem of Baire, so has the Baire property, is also called Baire space or Baire space. Certain phrases that apply to certain classes of topological spaces, the set of Baire, also subsumes the set of Baire.

The first versions of the sentence come from William Fogg Osgood (1897 derivation for the special case of the real line ) and René Louis Baire (1899 derivation for the special case of Euclidean space ).

• 3.1 Examples of Baire spaces

• 4.1 existence of nowhere differentiable functions
• 4.2 basis of a Banach space
• 4.3 Countable locally compact topological groups

Formulation for complete metric spaces

In a complete metric space, the set of Baire, which says there is true:

Baire categories

The concept of a meager amount is here For more general formulations and further characterizations first introduced.

Definition

Be a topological space (especially, for example, a metric space ).

• A subset is called meager or of first ( Baire ) category, if there is a countable set of nowhere dense subsets of whose union is all about.
• If a subset of non- first Baire category and is lean, then it is called of second ( Baire ) category ( or fat ).
• If the complement of a set is meager, it is called Residual or kommagers.

Examples of residual amounts of lean and

• The subset of the rational numbers is lean in, the set of irrational numbers is residually in.
• Is not lean in but lean in.
• The Cantor set is a non- countable, closed and nowhere dense, meager subset of.
• Each dense open subset is residually because its complement is nowhere dense. If this were not the case, it would have as a closed set a non- empty interior, so that the given open subset might not be tight.

Definition of a Baire space

A topological space is said to now exactly then bairesch meets or exceeds the set of Baire if every residual subset is dense in the space. This implies in particular that the space insofar as it is not empty, not even is lean, because its complement is the empty set, and this is tight in any non-empty topological space. In addition, this implies that no residual amount is lean, because since its complement is meager, the whole space as a union of two lean amounts would otherwise be lean.

It can be a topological space, the property of being a Baire space, characterized by one of the following conditions in an equivalent way:

• The union of a countable family closed subsets without interior points has no interior point.
• The average of a countable family of open, dense subsets is still tight in space.
• An open, non-empty subset is never lean.

Examples of Baire spaces

Applications

The set of Baire allows elegant proofs central propositions of classical functional analysis:

• Principle of uniform boundedness
• Banach - Steinhaus
• Set over the open mapping
• Set of Osgood ( Functional Analysis )

Existence of nowhere differentiable functions

On exist continuous functions that are differentiable at any point. To see this one uses for

Provides you the vector space with the supremum norm, then one can show that open and close in lies. Because the set of Baire we know that the space is dense in. The functions are continuous and differentiable at any point.

Basis of a Banach space

Another application of the theorem of Baire shows that every infinite-dimensional Banach space is based on an uncountable.

Evidence by the opposing assumption that there is a countable basis of the Banach space. Be. Then:

• As a finite dimensional vector spaces are complete,
• Their union gives the whole room.

By the theorem of Baire one of a sphere must contain. However, a subspace containing a ball, is always the whole room. This would lead to a finite-dimensional space, which leads to a contradiction.

Countable locally compact topological groups

With the set of Baire can be shown that at most countable locally compact, Hausdorff topological groups are discrete: they are the union of at most countably many singleton sets. These are complete, thus at least one of them shall be open at the set of Baire. That is, there are in the group an isolated point, but so are all points in isolation, as topological groups are homogeneous, and the discrete topology.

Similar conceptions in measure theory

In measure theory it is shown that the space provided with the Hausdorff and Lebesgue measure can not be written as a countable union of null sets. Substituting here the term null set by a meager amount obtained in this special case the statement of the Baire category theorem. The Baire categories can thus be seen as a topological analogue of null sets in measure theory and measure spaces. Indeed, there are many common features. These are extensively described in Oxtoby (1980).