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| {{For|the concept in set theory|Baire space (set theory)}}
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| {{more footnotes|date=March 2013}}
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| In [[mathematics]], a '''Baire space''' is a [[topological space]] which has "enough" points that any intersection of a [[countable]] collection of [[Open set|open]] [[dense set]]s in the space is also dense. [[Complete metric space]]s and [[locally compact]] [[Hausdorff spaces]] are examples of Baire spaces. The space is named in honor of [[René-Louis Baire]] who introduced the concept.
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| == Motivation ==
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| In an arbitrary topological space, the class of [[closed set]]s with [[empty set|empty]] [[interior (topology)|interior]] consists precisely of the [[boundary (topology)|boundaries]] of [[dense set|dense]] [[open set]]s. These sets are, in a certain sense, "negligible".
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| Some examples are finite sets in ℝ, smooth [[curve]]s in the plane, and proper [[affine space|affine subspaces]] in a [[Euclidean space]]. If a topological space is a Baire space then it is "large", meaning that it is not a [[countable]] [[union (set theory)|union]] of [[Negligible set|negligible subsets]]. For example, the three dimensional Euclidean space is not a countable union of its affine planes.
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| == Definition ==
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| 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.
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| === Modern definition ===
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| A topological space is called a '''Baire space''' if the [[union (set theory)|union]] of any [[countable]] [[Class (set theory)|collection]] of [[closed set]]s with [[empty set|empty]] [[interior (topology)|interior]] has empty interior.
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| This definition is equivalent to each of the following conditions:
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| * Every intersection of countably many [[dense set|dense]] [[open set]]s is dense.
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| * The interior of every [[union (set theory)|union]] of countably many [[closed set|closed]] [[nowhere dense]] sets is empty.
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| * Whenever the union of countably many closed subsets of ''X'' has an interior point, then one of the closed subsets must have an interior point.
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| === Historical definition ===<!-- This section is linked from [[Antiderivative]] -->
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| {{main|Meagre set}}
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| In his original definition, Baire defined a notion of category (unrelated to [[category theory]]) as follows.
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| A subset of a topological space ''X'' is called
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| * '''[[nowhere dense set|nowhere dense]]''' in ''X'' if the [[interior (topology)|interior]] of its [[closure (topology)|closure]] is [[empty set|empty]]
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| * of '''first category''' or '''[[meagre set|meagre]]''' in ''X'' if it is a union of countably many nowhere dense subsets
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| * of '''second category''' or '''nonmeagre''' in ''X'' if it is not of first category in ''X''
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| 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.
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| A subset ''A'' of ''X'' is '''comeagre''' if its [[Complement (set theory)|complement]] <math>X\setminus A</math> is meagre. A topological space ''X'' is a Baire space if and only if every comeager subset of ''X'' is dense.
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| === Choice of Terminology ===
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| ==== Nowhere/Somewhere/Everywhere Dense ====
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| One can define denseness in the following sense:<br>
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| A subset is dense at some point iff its closure is a neighborhood of that point. <math>\overline{A}\in\mathcal{N}_x</math><br>
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| One should compare this to the notion of closeness:<br>
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| A subset is close to some point iff that point is contained in its closure. <math>x\in\overline{A}</math><br>
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| So, similiraties become evident.
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| Now, this would give back precisely what is understood by a nowhere, somewhere and everywhere dense subset<br>
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| However, there's a flaw in there since neither the complement of nowhere dense subset is everywhere dense in general nor the complement of an everywhere dense subset is nowhere dense in general.
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| ''So one should pay attention or rather use a terminology not related to denseness!''
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| An example is given by.<ref>{{cite web|title=The Baire Category Theorem and its Consequences|url=http://www.ucl.ac.uk/~ucahad0/3103_handout_7.pdf}}</ref> A subset is called small iff the interior of its closure is empty and large iff the closure of its interior is all. Note that the complement of a small subset is large and vice versa. However, being not small resp. not large does not imply being large resp. small in general. But that happens for various topological notions as for being close resp. open too.
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| ==== Dense ====
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| Moreover, the same problem arises when considering the closure and dense subsets and their complement as well:<br>
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| The complement of a dense subset is, well, not dense?!<br>
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| This can be solved by rather considering the interior resp. exterior:<br>
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| In this sense, the interior of the complement is empty whenever the exterior is empty and vice versa.<br>
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| Dense subsets in this setting are precisely the ones with empty interior.<br>
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| Note also, the only subset whose interior is all is the space itself and its complement the empty set is the only subset whose exterior is all.<br>
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| Since being dense without the adverbials nowhere somewhere and everywhere is not contradicting itself, and since being dense is a commonly accepted notion, it would be beneficial to introduce a complementary notion especially in order to enlighten the interplay of Baire's Categories. This way in Baire Spaces, a countable intersection of large subsets (residual) is dense while analogously a countable union of small subsets (meagre) is complementary to being dense that is "sparse" in some sense.
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| ==== Meagre/Residual ====
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| While meagre seems intuitive its complementary notion is rather confusing: Residual!<br>
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| A residual subset is one given by countable intersection of large subsets that is still big in some sense in Baire Spaces, namely dense. Therefore residual becomes counterintuitive. One should rather call it fat or at least adipose when following Baire's terminology.
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| == Examples ==
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| * The space '''R''' of [[real number]]s with the usual topology, is a Baire space, and so is of second category in itself. The [[rational numbers]] are of first category and the [[irrational numbers]] are of second category in '''R'''.
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| * The [[Cantor set]] is a Baire space, and so is of second category in itself, but it is of first category in the interval [0, 1] with the usual topology.
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| * Here is an example of a set of second category in '''R''' with [[Lebesgue measure]] 0.
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| ::<math>\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} \left(r_{n}-{1 \over 2^{n+m} }, r_{n}+{1 \over 2^{n+m}}\right)</math>
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| :where <math> \left\{r_{n}\right\}_{n=1}^{\infty} </math> is a [[sequence]] that [[Countable|enumerates]] the [[rational number]]s.
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| * Note that the space of [[rational number]]s with the usual topology inherited from the [[real number|reals]] is not a Baire space, since it is the union of countably many closed sets without interior, the [[singleton (mathematics)|singleton]]s.
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| == Baire category theorem ==
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| {{main|Baire category theorem}}
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| The [[Baire category theorem]] gives [[sufficient condition]]s for a topological space to be a Baire space. It is an important tool in [[topology]] and [[functional analysis]].
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| *('''BCT1''') Every [[complete space|complete]] [[metric space]] is a Baire space. More generally, every topological space which is [[homeomorphic]] to an [[open set|open subset]] of a [[complete space|complete]] [[pseudometric space]] is a Baire space. In particular, every [[completely metrizable]] space is a Baire space.
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| *('''BCT2''') Every [[locally compact]] [[Hausdorff space]] is a Baire space.
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| '''BCT1''' shows that each of the following is a Baire space:
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| * The space '''R''' of [[real number]]s
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| * The space of [[irrational number]]s
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| * The [[Cantor set]]
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| * Indeed, every [[Polish space]]
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| '''BCT2''' shows that every [[manifold]] is a Baire space, even if it is not [[paracompact]], and hence not [[metrizable]]. For example, the [[long line (topology)|long line]] is of second category.
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| == Properties ==
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| *Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of ''X'' is non-empty, but the converse of neither of these is true, as is shown by the [[topological disjoint sum]] of the rationals and the [[unit interval]] [0, 1].
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| *Every [[open subspace (topology)|open subspace]] of a Baire space is a Baire space.
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| *Given a [[indexed family|family]] of [[continuous function (topology)|continuous]] functions ''f''<sub>''n''</sub>:''X''→''Y'' with pointwise limit ''f'':''X''→''Y''. If ''X'' is a Baire space then the points where ''f'' is not continuous is ''a [[meagre set]]'' in ''X'' and the set of points where ''f'' is continuous is dense in ''X''. A special case of this is the [[uniform boundedness principle]].
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| ==See also==
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| * [[Banach–Mazur game]]
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| * [[Descriptive set theory]]
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| * [[Baire space (set theory)]]
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| ==References==
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| {{reflist}}
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| ==Sources==
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| *Munkres, James, ''Topology'', 2nd edition, Prentice Hall, 2000.
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| *Baire, René-Louis (1899), Sur les fonctions de variables réelles, ''Annali di Mat. Ser. 3'' '''3''', 1--123.
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Baire_space Encyclopaedia of Mathematics article on Baire space]
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| * [http://www.encyclopediaofmath.org/index.php/Baire_theorem Encyclopaedia of Mathematics article on Baire theorem]
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| [[Category:General topology]]
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| [[Category:Functional analysis]]
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| [[Category:Properties of topological spaces]]
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At the time a association struggle begins, you will see Often the particular War Map, a good solid map of this gua area area association battles booty place. Welcoming territories will consistently be on the left, while having the adversary association inside of the right. boondocks anteroom on some sort of war map represents some kind of war base.
Transferring from band blueprint toward a besprinkle blueprint induces some sort of added in authentic picture. The actual accumbent time arbor will be actually scaled evenly. But it's adamantine turn out to be able to acquaint what's activity now inside bottom-left bend now. The main ethics are so bunched up you simply will not acquaint them afar any longer.
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