About: Final topology

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In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous. The dual notion is the initial topology, which for a given family of functions from a set into topological spaces is the coarsest topology on that makes those functions continuous.

Attributes	Values
rdfs:label	Finaltopologie (de) Topologie finale (fr) Final topology (en) Topologia finale (it) Finale topologie (nl)
rdfs:comment	Als Finaltopologie bezüglich einer Abbildungsfamilie bezeichnet man im mathematischen Teilgebiet der Topologie die feinste Topologie auf einer Menge , die diese Familie von Abbildungen aus anderen topologischen Räumen nach stetig macht. Die Finaltopologie entsteht also durch „Vorwärtsübertragung“ der auf den Urbildräumen vorhandenen topologischen Strukturen auf die Menge . Dies ist die Anwendung eines allgemeineren Konzepts aus der Kategorientheorie auf topologische Räume, mit der wichtige „natürliche Räume“ wie Quotienten- und Summenräume in einen gemeinsamen Rahmen gestellt werden können. Je nach Kontext spricht man dann auch von Quotiententopologie bzw. Summentopologie. (de) En mathématiques et plus précisément en topologie, la topologie finale, sur un ensemble d'arrivée commun à une famille d'applications définies chacune sur un espace topologique, est la topologie la plus fine pour laquelle toutes ces applications sont continues. La notion duale est celle de topologie initiale. (fr) In matematica, in particolare in topologia generale, la topologia finale o topologia forte su un insieme rispetto ad una famiglia di funzioni è la topologia più fine tale per cui le funzioni della famiglia sono continue. La struttura duale alla topologia finale è detta topologia iniziale. (it) In de topologie, een tak van de wiskunde, is de finale topologie op een verzameling met betrekking tot een collectie afbeeldingen naar die verzameling, de fijnste topologische structuur die deze afbeeldingen continu maakt. (nl) In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous. The dual notion is the initial topology, which for a given family of functions from a set into topological spaces is the coarsest topology on that makes those functions continuous. (en)
foaf:depiction
dcterms:subject	General topology
Wikipage page ID	1819736 (xsd:integer)
Wikipage revision ID	1070273451 (xsd:integer)
Link from a Wikipage to another Wikipage	Product topology Metrizable space Topological embedding Inclusion map Initial topology LB-space Complete topological vector space Mathematics General topology Fréchet–Urysohn space Locally convex topological vector space Comma category Compactly generated space Comparison of topologies Étalé space Functor Functor category Arc (topology) Adjoint functors Category of sets Topological vector space Disjoint union (topology) Hausdorff space Locally compact space Euclidean space Diagonal functor Direct limit Discrete category Family of sets Forgetful functor Quotient space (topology) Intersection (set theory) First-countable General topology Bijection Coherent topology Discrete topology Disjoint union Manifold Continuous function (topology) If and only if Natural number Category of topological spaces Category theory Sequential space Set (mathematics) Infimum Lattice of topologies Euclidean topology Disjoint union topology Image (mathematics) Subspace topology First-countable space Topological space Canonical injection Fréchet-Urysohn space Finest topology Category of co-cones Direct system Direct system (mathematics) Quotient map Quotient topology Path (mathematics) Singleton set Coarsest topology Space of all real sequences Space of finite sequences Locally path-connected Surjective map
Link from a Wikipage to an external page	https://archive.org/details/generaltopology00will_0
sameAs	Final topology Final topology Final topology Final topology Final topology Final topology Final topology Final topology
dbp:wikiPageUsesTemplate	dbt:Annotated_link dbt:Cite_book dbt:Em dbt:Reflist dbt:Short_description dbt:Willard_General_Topology
thumbnail	wiki-commons:Special:FilePath/FinalTopology-01.png?width=300

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