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en>Marcs2001: changed 'If atomic factors did have' to 'If atomic factors did not have'

2014-06-27T08:15:19Z

changed 'If atomic factors did have' to 'If atomic factors did not have'

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	~~{{For\|the concept in topology\|Baire space}}~~		Wilber Berryhill is what his wife loves to call him and he totally loves this title. My spouse and I reside in Mississippi but now I'm contemplating other choices. To perform lacross is the factor I love most of all. Office supervising is where her primary income comes from.<br><br>My blog post ... free psychic reading - [http://www.comictube.com/users/INevile www.comictube.com try these out],

	~~In [[set theory]], the '''Baire space'''~~ is ~~the [[Set (mathematics)\|set]] of all [[infinite sequence]]s of [[natural number]]s with a certain [[topology]]. This space is commonly used in [[descriptive set theory]],~~ to the extent that its elements are often called “reals.” It is often denoted '''B''', '''N'''<sup>'''N'''</sup>, ω<sup>ω</sup>, or <sup>ω</sup>ω. [[Yiannis N. Moschovakis\|Moschovakis]] denotes it <math>\mathcal{N}</math>.

	~~The Baire space is defined to be the [[Cartesian product]] of [[countable set\|countably infinitely]] many copies of the set of natural numbers,~~ and is given the [[product topology]] (where each copy of the set of natural numbers is given the [[discrete topology]]). The Baire space is often represented using the [[tree (descriptive set theory)\|tree]] of finite sequences of natural numbers.

	~~The Baire space can be contrasted with [[Cantor space]], the set of infinite sequences of [[binary digit]]s~~.

	~~== Topology~~ and ~~trees ==~~

	~~The product topology used to define the Baire space can be described more concretely~~ in ~~terms of trees. The definition of the product topology leads to this characterization of [[base (topology)\|basic open sets]]:~~
	:If any finite set of natural number coordinates {''c''<sub>''i''</sub> : ''i'' < ''n'' } is selected, and for each ''c''<sub>''i''</sub> a particular natural number value ''v''<sub>''i''</sub> is selected, then the set of all infinite sequences of natural numbers that have value ''v''<sub>''i''</sub> at position ''c''<sub>''i''~~</sub> for all ''i'' < ''n'' is a basic open set~~. ~~Every open set~~ is ~~a union of a collection of these.~~

	~~By moving to a different basis for~~ the ~~same topology, an alternate characterization of open sets can be obtained:~~
	~~:If a sequence~~ of ~~natural numbers {''w''<sub>''i''</sub> : ''i'' < ''n''} is selected, then the set of all infinite sequences of natural numbers that have value ''w''<sub>''i''</sub> at position ''i'' for~~ all ~~''i'' < ''n'' is a basic open set~~. ~~Every open set~~ is ~~a union of a collection of these~~.

	Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers, and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space as the set of all paths through the full tree ω<~~sup~~><~~ω</sup~~> ~~of finite sequences of natural numbers ordered by extension~~. ~~An open set is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set if and only if its path goes through one of these nodes~~.

	~~The representation of the Baire space as paths through a tree also gives a characterization of closed sets~~. ~~For any closed subset ''C'' of Baire space there is a subtree ''T'' of ω<sup><ω<~~/~~sup> such that any point ''x'' is in ''C'' if and only if ''x'' is a path through ''T''. Conversely, the set of paths through any subtree of ω<sup><ω<~~/~~sup> is a closed set~~.

	~~== Properties ==~~
	~~The Baire space has the following properties:~~

	~~# It is a [[perfect set\|perfect]] [[Polish space]], which means it is a [[complete metric space\|completely metrizable ]] [[second countable]] space with no [[isolated point]]s~~. ~~As such, it has the same [[cardinality]] as the real line and is a [[Baire space]] in the topological sense of the term.~~
	~~# It is [[zero dimensional]] and [[totally disconnected]].~~
	~~# It is not [[locally compact]].~~
	~~# It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space. Moreover, any Polish space has a [[dense set\|dense]] [[Gδ set\|G<sub>δ<~~/~~sub>]] subspace [[homeomorphic]] to a G<sub>δ<~~/~~sub> subspace of the Baire space~~.
	~~# The Baire space is homeomorphic to the product of any finite or countable number of copies of itself~~.

	~~== Relation to the real line ==~~
	The Baire space is [[homeomorphic]] to the set of [[irrational number]]s when they are given the [[subspace topology]] inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using [[continued fraction]]s.

	From the point of view of [[descriptive set theory]], the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it is often possible to prove results about arbitrary Polish spaces by showing that these ~~properties hold for Baire space and by showing that they are preserved by [[continuous functions]].~~

	~~'''B''' is also of independent, but minor, interest in [[real analysis]~~], where it is considered as a [[uniform space]]. The uniform structures of '''B''' and '''Ir''' (the irrationals) are different, however: '''B''' is [[complete space\|complete]] in its usual metric while '''Ir''' is not (although these spaces are homeomorphic).

	~~== References ==~~
	* {{cite book \|authorlink=Alexander S. Kechris\| author=Kechris, Alexander S. \| title=Classical Descriptive Set Theory \| publisher=Springer-Verlag \| year=1994 \| isbn=0-387-94374-9}}
	* {{cite book \|authorlink=Yiannis N. Moschovakis\| author=Moschovakis, Yiannis N. \| title=Descriptive Set Theory \| publisher=North Holland \| year=1980 \|isbn=0-444-70199-0}}

	~~[[Category:Descriptive set theory]]~~

en>Yobot: /* References */clean up /fixed checkwiki error 18 using AWB (8717)

2012-12-01T15:06:39Z

References: clean up /fixed checkwiki error 18 using AWB (8717)

← Older revision		Revision as of 17:06, 1 December 2012
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	I'~~m Rosemary and I live in~~ a ~~seaside city~~ in ~~northern Australia~~, ~~Maroon~~. I'~~m 34 and I~~'~~m will soon finish my study at Comparative Politics~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my page; FIFA 15 coin hack -~~ [~~http~~://~~starinmotion~~.~~com~~/~~index~~.~~php?do=~~/~~profile-123922~~/~~info~~/~~info~~/~~info~~/~~info/ visit~~ the ~~following website page~~] -		{{For\|the concept in topology\|Baire space}}

			In [[set theory]], the '''Baire space''' is the [[Set (mathematics)\|set]] of all [[infinite sequence]]s of [[natural number]]s with a certain [[topology]]. This space is commonly used in [[descriptive set theory]], to the extent that its elements are often called “reals.” It is often denoted '''B''', '''N'''<sup>'''N'''</sup>, ω<sup>ω</sup>, or <sup>ω</sup>ω. [[Yiannis N. Moschovakis\|Moschovakis]] denotes it <math>\mathcal{N}</math>.

			The Baire space is defined to be the [[Cartesian product]] of [[countable set\|countably infinitely]] many copies of the set of natural numbers, and is given the [[product topology]] (where each copy of the set of natural numbers is given the [[discrete topology]]). The Baire space is often represented using the [[tree (descriptive set theory)\|tree]] of finite sequences of natural numbers.

			The Baire space can be contrasted with [[Cantor space]], the set of infinite sequences of [[binary digit]]s.

			== Topology and trees ==

			The product topology used to define the Baire space can be described more concretely in terms of trees. The definition of the product topology leads to this characterization of [[base (topology)\|basic open sets]]:
			:If any finite set of natural number coordinates {''c''<sub>''i''</sub> : ''i'' < ''n'' } is selected, and for each ''c''<sub>''i''</sub> a particular natural number value ''v''<sub>''i''</sub> is selected, then the set of all infinite sequences of natural numbers that have value ''v''<sub>''i''</sub> at position ''c''<sub>''i''</sub> for all ''i'' < ''n'' is a basic open set. Every open set is a union of a collection of these.

			By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:
			:If a sequence of natural numbers {''w''<sub>''i''</sub> : ''i'' < ''n''} is selected, then the set of all infinite sequences of natural numbers that have value ''w''<sub>''i''</sub> at position ''i'' for all ''i'' < ''n'' is a basic open set. Every open set is a union of a collection of these.

			Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers, and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space as the set of all paths through the full tree ω<sup><ω</sup> of finite sequences of natural numbers ordered by extension. An open set is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set if and only if its path goes through one of these nodes.

			The representation of the Baire space as paths through a tree also gives a characterization of closed sets. For any closed subset ''C'' of Baire space there is a subtree ''T'' of ω<sup><ω</sup> such that any point ''x'' is in ''C'' if and only if ''x'' is a path through ''T''. Conversely, the set of paths through any subtree of ω<sup><ω</sup> is a closed set.

			== Properties ==
			The Baire space has the following properties:

			# It is a [[perfect set\|perfect]] [[Polish space]], which means it is a [[complete metric space\|completely metrizable ]] [[second countable]] space with no [[isolated point]]s. As such, it has the same [[cardinality]] as the real line and is a [[Baire space]] in the topological sense of the term.
			# It is [[zero dimensional]] and [[totally disconnected]].
			# It is not [[locally compact]].
			# It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space. Moreover, any Polish space has a [[dense set\|dense]] [[Gδ set\|G<sub>δ</sub>]] subspace [[homeomorphic]] to a G<sub>δ</sub> subspace of the Baire space.
			# The Baire space is homeomorphic to the product of any finite or countable number of copies of itself.

			== Relation to the real line ==
			The Baire space is [[homeomorphic]] to the set of [[irrational number]]s when they are given the [[subspace topology]] inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using [[continued fraction]]s.

			From the point of view of [[descriptive set theory]], the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and by showing that they are preserved by [[continuous functions]].

			'''B''' is also of independent, but minor, interest in [[real analysis]], where it is considered as a [[uniform space]]. The uniform structures of '''B''' and '''Ir''' (the irrationals) are different, however: '''B''' is [[complete space\|complete]] in its usual metric while '''Ir''' is not (although these spaces are homeomorphic).

			== References ==
			* {{cite book \|authorlink=Alexander S. Kechris\| author=Kechris, Alexander S. \| title=Classical Descriptive Set Theory \| publisher=Springer-Verlag \| year=1994 \| isbn=0-387-94374-9}}
			* {{cite book \|authorlink=Yiannis N. Moschovakis\| author=Moschovakis, Yiannis N. \| title=Descriptive Set Theory \| publisher=North Holland \| year=1980 \|isbn=0-444-70199-0}}

			[[Category:Descriptive set theory]]

en>Open2universe: link repair (You can help!)

2009-07-01T00:46:23Z

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