Symplectic vector field - Revision history

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Added some explanation to the prose and wikilinks, also c/e

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	~~In [[general topology]]~~ and ~~related areas of [[mathematics]],~~ the '~~''final topology''' (or '''strong topology''' or '''colimit topology''' or '''projective topology''') on~~ a ~~[[Set (mathematics)\|set]] <math>X</math>, with respect to~~ a ~~family~~ of ~~functions into <math>X</math>~~, ~~is the~~ [~~[finest topology]] on ''X'' which makes those functions [[continuous function (topology)\|continuous]]~~.		Another day I woke up and noticed - At the moment I've been solitary for a while and after much intimidation from buddies I today locate myself registered for internet dating. They promised me that there are a lot of standard, [http://www.Adobe.com/cfusion/search/index.cfm?term=&pleasant&loc=en_us&siteSection=home pleasant] and interesting people to meet, therefore the pitch is gone by here!<br>My buddies and fam are awe-inspiring and spending some time together at bar gigabytes or dinners is obviously a necessity. As I discover that one may never own a significant dialogue together with the sound I haven't ever been into nightclubs. I additionally have two cheap Luke Bryan Tickets ([http://www.museodecarruajes.org www.Museodecarruajes.org]) definitely cheeky and really adorable canines who are consistently ready to meet up new individuals.<br>luke bryan tour dates 2014 ([http://www.banburycrossonline.com visit our website]) I make an effort to keep as toned as possible staying at the gymnasium [http://lukebryantickets.neodga.com luke brice] several times a week. I love my athletics and attempt to play or see while many a potential. I am going to frequently at Hawthorn matches being wintertime. Note: If you contemplated shopping a sport I don't mind, I've experienced the carnage of fumbling luke bryan tickets for sale ([http://lukebryantickets.iczmpbangladesh.org iczmpbangladesh.org]) fits at stocktake revenue.<br><br><br><br>Feel free to surf to my blog post: find luke bryan tickets ([http://www.hotelsedinburgh.org http://www.hotelsedinburgh.org])

	~~The dual notion is the [[initial topology]]~~.

	== ~~Definition~~ ==

	~~Given a set <math>X</math>~~ and ~~a family of [[topological space]]s <math>Y_i</math> with functions~~
	~~:<math>f_i: Y_i \~~to ~~X</math>~~
	the ~~'''final topology''' <math>\tau</math> on <math>X</math>~~ is ~~the [[finest topology]] such that each~~
	:<~~math~~>~~f_i: Y_i \to (X,\tau)</math>~~
	is ~~[[continuous function (topology)\|continuous]]~~.

	~~Explicitly, the final topology~~ may ~~be described as follows:~~ a ~~subset~~ '~~'U'' of ''X'' is open~~ [~~[if and only if]] <math>f_i^{-1}(U)<~~/~~math> is open in ''Y''<sub>''i''<~~/~~sub> for each ''i'' ∈ ''I''~~.

	~~== Examples ==~~

	* The [[quotient topology]] is the final topology on the quotient space with respect to the [[quotient map]].
	* The [[disjoint union (topology)\|disjoint union]] is the final topology with respect to the family of [[canonical injection]]s.
	* More generally, a topological space is [[coherent topology\|coherent]] with a family of subspaces if it has the final topology coinduced by the inclusion maps.
	* The [[direct limit]] ~~of any [[direct system (mathematics~~)~~\|direct system]] of spaces~~ and ~~continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms~~.
	* Given a [[family of sets\|family]] of topologies {τ<~~sub~~>~~''i''<~~/~~sub>} on a fixed set ''X'' the final topology on ''X'' with respect~~ to the ~~functions id<sub>''X''</sub>~~ : ~~(''X'', τ<sub>''i''<~~/~~sub>) → ''X'' is the [[infimum]] (or meet) of the topologies {τ<sub>''i''<~~/~~sub>} in the [[lattice of topologies]] on ''X''~~. ~~That is, the final topology τ is the [[intersection (set theory)\|intersection]] of the topologies {τ<sub>''i''</sub>}~~.
	* The [[etale space]] of a ~~sheaf is topologized by~~ a ~~final topology~~.

	~~== Properties ==~~

	~~A subset of <math>X</math> is closed/open [[if and only if]] its preimage under ''f''<sub>''i''</sub> is closed/open in <math>Y_i</math> for each ''i'' ∈ ''~~I''.

	~~The final topology on ''X'' can be characterized by the following [[universal property]]~~: a ~~function <math>g</math> from <math>X</math> to some space <math>Z</math> is continuous if and only if <math>g \circ f_i</math> is continuous for each ''i'' ∈ '~~'I''.
	~~[[Image:FinalTopology-01.png\|center\|Characteristic property of the final topology]]~~

	By the ~~universal property~~ of ~~the [~~[~~disjoint union topology]] we know that given any family of continuous maps ''f''<sub>''i''</sub>~~ : ~~''Y''<sub>''i''<~~/~~sub> → ''X'' there is a unique continuous map~~
	~~:<math>f\colon \coprod_i Y_i \to X</math>~~
	~~If the family of maps ''f''<sub>''i''<~~/~~sub> ''covers'' ''X'' (i~~.e. each ''x'' in ''X'' lies in the image of some ''f''<sub>''i''</sub>) then the map ''f'' will be a [[quotient map]] if and only if ''X'' has the final topology determined by the maps ''f''<sub>''i''</sub>.

	~~== Categorical description ==~~

	~~In the language of [[category theory~~]~~], the final topology construction can be described as follows~~. ~~Let ''Y'' be a [[functor]] from a [[discrete category]] ''J'' to the [[category of topological spaces]] '''Top''' which selects the spaces ''Y''~~<~~sub~~>~~''i''~~<~~/sub~~> ~~for ''i'' in ''J''. Let Δ be the [[diagonal functor]] from '''Top''' to the [[functor category]] '''Top'''~~<~~sup~~>~~''J''~~<~~/sup~~> ~~(this functor sends each space ''X''~~ to ~~the constant functor~~ to ~~''X''). The [[comma category]]~~ (~~''Y'' ↓ Δ) is then the~~ [~~[category of cones]] from ''Y'', i~~.e. ~~objects in (''Y'' ↓ Δ) are pairs (''X'', ''f'') where ''f''<sub>''i''</sub>~~ : ~~''Y''<sub>''i''<~~/sub> → ''X'' is a family of continuous maps to ''X''. If ''U'' is the [[forgetful functor]] from '''Top''' to '''Set''' and Δ′ is the diagonal functor from '''Set''' to '''Set'''<sup>''J''</sup> then the comma category (''UY'' ↓ Δ′) is the category of all cones from ''UY''. The final topology construction can then be described as a functor from (''UY'' ↓ Δ′) to (''Y'' ↓ Δ). ~~This functor is [[adjoint functors\|left adjoint]] to the corresponding forgetful functor~~.

	~~== See also ==~~
	* [[Initial topology]]

	~~==References==~~
	* Stephen Willard, ''General Topology'', (1970) ~~Addison-Wesley Publishing Company, Reading Massachusetts. ''(Provides a short, general introduction)''~~

	~~[[Category:General topology]]~~

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2013-03-23T10:14:34Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7661853

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	~~Graphic Designer Garry from Windsor~~, ~~usually spends time~~ with ~~pursuits~~ which ~~include warships~~, ~~free coins fifa 14 hack~~ and ~~architecture~~. ~~Identified~~ some ~~lovely spots after working 2 weeks at Messel Pit Fossil Site.~~<br><br>~~My web page fifa 14 hack apk~~ - [~~https~~://~~www~~.~~youtube~~.~~com~~/~~watch?v~~=~~HSoNAvTIeEc simply click~~ the ~~up coming internet site~~] -		In [[general topology]] and related areas of [[mathematics]], the '''final topology''' (or '''strong topology''' or '''colimit topology''' or '''projective topology''') on a [[Set (mathematics)\|set]] <math>X</math>, with respect to a family of functions into <math>X</math>, is the [[finest topology]] on ''X'' which makes those functions [[continuous function (topology)\|continuous]].

			The dual notion is the [[initial topology]].

			== Definition ==

			Given a set <math>X</math> and a family of [[topological space]]s <math>Y_i</math> with functions
			:<math>f_i: Y_i \to X</math>
			the '''final topology''' <math>\tau</math> on <math>X</math> is the [[finest topology]] such that each
			:<math>f_i: Y_i \to (X,\tau)</math>
			is [[continuous function (topology)\|continuous]].

			Explicitly, the final topology may be described as follows: a subset ''U'' of ''X'' is open [[if and only if]] <math>f_i^{-1}(U)</math> is open in ''Y''<sub>''i''</sub> for each ''i'' ∈ ''I''.

			== Examples ==

			* The [[quotient topology]] is the final topology on the quotient space with respect to the [[quotient map]].
			* The [[disjoint union (topology)\|disjoint union]] is the final topology with respect to the family of [[canonical injection]]s.
			* More generally, a topological space is [[coherent topology\|coherent]] with a family of subspaces if it has the final topology coinduced by the inclusion maps.
			* The [[direct limit]] of any [[direct system (mathematics)\|direct system]] of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms.
			* Given a [[family of sets\|family]] of topologies {τ<sub>''i''</sub>} on a fixed set ''X'' the final topology on ''X'' with respect to the functions id<sub>''X''</sub> : (''X'', τ<sub>''i''</sub>) → ''X'' is the [[infimum]] (or meet) of the topologies {τ<sub>''i''</sub>} in the [[lattice of topologies]] on ''X''. That is, the final topology τ is the [[intersection (set theory)\|intersection]] of the topologies {τ<sub>''i''</sub>}.
			* The [[etale space]] of a sheaf is topologized by a final topology.

			== Properties ==

			A subset of <math>X</math> is closed/open [[if and only if]] its preimage under ''f''<sub>''i''</sub> is closed/open in <math>Y_i</math> for each ''i'' ∈ ''I''.

			The final topology on ''X'' can be characterized by the following [[universal property]]: a function <math>g</math> from <math>X</math> to some space <math>Z</math> is continuous if and only if <math>g \circ f_i</math> is continuous for each ''i'' ∈ ''I''.
			[[Image:FinalTopology-01.png\|center\|Characteristic property of the final topology]]

			By the universal property of the [[disjoint union topology]] we know that given any family of continuous maps ''f''<sub>''i''</sub> : ''Y''<sub>''i''</sub> → ''X'' there is a unique continuous map
			:<math>f\colon \coprod_i Y_i \to X</math>
			If the family of maps ''f''<sub>''i''</sub> ''covers'' ''X'' (i.e. each ''x'' in ''X'' lies in the image of some ''f''<sub>''i''</sub>) then the map ''f'' will be a [[quotient map]] if and only if ''X'' has the final topology determined by the maps ''f''<sub>''i''</sub>.

			== Categorical description ==

			In the language of [[category theory]], the final topology construction can be described as follows. Let ''Y'' be a [[functor]] from a [[discrete category]] ''J'' to the [[category of topological spaces]] '''Top''' which selects the spaces ''Y''<sub>''i''</sub> for ''i'' in ''J''. Let Δ be the [[diagonal functor]] from '''Top''' to the [[functor category]] '''Top'''<sup>''J''</sup> (this functor sends each space ''X'' to the constant functor to ''X''). The [[comma category]] (''Y'' ↓ Δ) is then the [[category of cones]] from ''Y'', i.e. objects in (''Y'' ↓ Δ) are pairs (''X'', ''f'') where ''f''<sub>''i''</sub> : ''Y''<sub>''i''</sub> → ''X'' is a family of continuous maps to ''X''. If ''U'' is the [[forgetful functor]] from '''Top''' to '''Set''' and Δ′ is the diagonal functor from '''Set''' to '''Set'''<sup>''J''</sup> then the comma category (''UY'' ↓ Δ′) is the category of all cones from ''UY''. The final topology construction can then be described as a functor from (''UY'' ↓ Δ′) to (''Y'' ↓ Δ). This functor is [[adjoint functors\|left adjoint]] to the corresponding forgetful functor.

			== See also ==
			* [[Initial topology]]

			==References==
			* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. ''(Provides a short, general introduction)''

			[[Category:General topology]]

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