|
|
| Line 1: |
Line 1: |
| 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]]
| |
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, pleasant and interesting people to meet, therefore the pitch is gone by here!
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 (www.Museodecarruajes.org) definitely cheeky and really adorable canines who are consistently ready to meet up new individuals.
luke bryan tour dates 2014 (visit our website) I make an effort to keep as toned as possible staying at the gymnasium 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 (iczmpbangladesh.org) fits at stocktake revenue.
Feel free to surf to my blog post: find luke bryan tickets (http://www.hotelsedinburgh.org)