The Baire category theorem is an important tool 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. In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ...

Statement of the theorem

• (BCT1) Every non-empty 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. In particular, every topologically complete space is a Baire space.
• (BCT2) Every non-empty locally compact Hausdorff space is a Baire space.

Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the Baire space of irrational numbers), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... This word should not be confused with homomorphism. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. ...

Relation to AC

The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact the statement that every complete pseudometric space is a Baire space is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice. [1] In mathematics, the axiom of choice is an axiom of set theory. ... In logic, statements p and q are logically equivalent if they have the same logical content. ... In mathematics, the axiom of dependent choice is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. ...

Uses of the theorem

BCT1 is used to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle. In mathematics, there are two theorems with the name open mapping theorem. Functional analysis In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y... In mathematics, the closed graph theorem is a basic result of functional analysis. ... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ...

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. In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... In mathematics, an uncountable set is a set which is not countable. ... Generally, a singleton is something which exists alone in some way. ... In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

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

• The space R of real numbers
• The Cantor set
• Every manifold (it is locally compact)
• Every topological space homeomorphic to a Baire space (e.g. the space of irrational numbers that is not complete with the distance coming from R)

Various applications of BCT1 and its relations with similar phenomenons are listed in the Bwatabaire (mostly in french but english is allowed) In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

References

• Schechter, Eric, Handbook of Analysis and its Foundations, Academic Press, ISBN 0-126-22760-8
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).