Baire room (general)

A Baire space , also called Baire space , is a special topological space in topology , a branch of mathematics . Baire spaces are named after René Louis Baire and have certain regularity properties. From a topological point of view they are large in the sense that they are not lean and therefore cannot be written anywhere as a countable union of dense sets .

There are many far-reaching implications in Baire spaces, especially for functional analysis . The Banach-Steinhaus theorem , the principle of uniform limitation and the theorem on open mapping can be derived from the fact that every complete metric space is a Baire space.

definition

A topological space is given . A crowd is nowhere called dense when the inside of its closure is empty. Furthermore, a set is called lean if it is the union of countably many nowhere dense sets. ${\displaystyle (X,{\mathcal {O}})}$

The topological space is now called a Baire space if one of the following equivalent conditions is met: ${\displaystyle X}$

(a) The complement of any lean set is dense in . ${\displaystyle X}$

(b) A non-empty open subset of is never lean. ${\displaystyle X}$

(c) Every union of at most countably many closed subsets of without internal points is itself without internal points. ${\displaystyle X}$

(d) Every cut of at most countably many open, in dense subsets is again dense in . ${\displaystyle X}$ ${\displaystyle X}$

There are also different names. So complements are of meager amounts also komagere amounts called meager amounts as sets of the first category and not lean amounts than amounts the second category referred.

Examples and characteristics

• Every completely metrizable room and therefore every Polish room is a Baire room. Most authors call this statement Baire's Theorem . In particular, the special baire room ℕ ^{ℕ is} a baire room.
• Likewise, every locally compact Hausdorff room is a Baire room. This statement is also referred to by some authors as Baire's theorem or subsumed under him.
• Every non-empty, open set of a Baire space, provided with the subspace topology , is again a Baire space. Likewise, in a compact Hausdorff space, every Gδ set is again a Baire space.
• According to Banach's category theorem, every topological space is a Baire space except for a meager amount . In addition, in a (non-empty) baire space, the complement of any lean set is again a baire space.
• If there is a countable covering of this space with closed sets in a non-empty Baire space, at least one of these sets has a non-empty interior. This statement forms the basis for the proof of the principle of uniform restriction .

See also

• Baire room (special)
• Baire's Theorem

literature

• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag, Lemgo 2006, ISBN 3-88538-115-X , p. 189 ff . ( MR2172813 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 , pp. 132 ff . ( MR0423277 ).
• Stephen Willard: General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA (et al.) 1970, pp. 185 ff . ( MR0264581 ).

Web links

• PS Aleksandrov: Baire space . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Baire Space . In: MathWorld (English).

Individual evidence

1. ^ ^{A b} Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin / Heidelberg 2012, ISBN 978-3-642-22260-3 , p.  229-333 , doi : 10.1007 / 978-3-642-22261-0 .
2. ↑ ^{a b c} Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin / Heidelberg / New York 2001, ISBN 978-3-540-67790-1 , p.  174-176 , doi : 10.1007 / 978-3-642-56860-2 .
3. ↑ Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg / Dordrecht / London / New York 2011, ISBN 978-3-642-21016-7 , pp.  139 , doi : 10.1007 / 978-3-642-21017-4 .
4. ↑ Horst Schubert: Topology. 1975, p. 134
5. ↑ Stephen Willard: General Topology. 1978, p. 186
6. ^ John C. Oxtoby: Measure and Category. A Survey of the Analogies between topological and measure spaces (= Graduate Texts in Mathematics. 2). 2nd edition. Springer, New York NY a. a. 1980, ISBN 3-540-90508-1 , p. 62