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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]].
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| The dual notion is the [[initial topology]].
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| == Definition ==
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| Given a set <math>X</math> and a family of [[topological space]]s <math>Y_i</math> with functions
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| :<math>f_i: Y_i \to X</math>
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| the '''final topology''' <math>\tau</math> on <math>X</math> is the [[finest topology]] such that each
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| :<math>f_i: Y_i \to (X,\tau)</math>
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| is [[continuous function (topology)|continuous]].
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| 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''.
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| == Examples ==
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| * The [[quotient topology]] is the final topology on the quotient space with respect to the [[quotient map]].
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| * The [[disjoint union (topology)|disjoint union]] is the final topology with respect to the family of [[canonical injection]]s.
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| * 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.
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| * 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.
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| * 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>}.
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| * The [[etale space]] of a sheaf is topologized by a final topology.
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| == Properties ==
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| 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''.
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| 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''.
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| [[Image:FinalTopology-01.png|center|Characteristic property of the final topology]]
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| 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
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| :<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>.
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| == Categorical description == | |
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| 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.
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| == See also ==
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| * [[Initial topology]]
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| ==References==
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| * Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. ''(Provides a short, general introduction)''
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| [[Category:General topology]]
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Hello!
My name is Selina and I'm a 25 years old girl from Dog Swamp.
My blog post - FIFA coin generator