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- Natural_topology abstract "In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that the topology in question arises naturally or canonically (see mathematical jargon) in the given context.Note that in some cases multiple topologies seem "natural". For example, if Y is a subset of a totally ordered set X, then the induced order topology, i.e. the order topology of the totally ordered Y, where this order is inherited from X, is coarser than the subspace topology of the order topology of X."Natural topology" does quite often have a more specific meaning, at least given some prior contextual information: the natural topology is a topology which makes a natural map or collection of maps continuous. This is still imprecise, even once one has specified what the natural maps are, because there may be many topologies with the required property. However, there is often a finest or coarsest topology which makes the given maps continuous, in which case these are obvious candidates for the natural topology.The simplest cases (which nevertheless cover many examples) are the initial topology and the final topology (Willard (1970)). The initial topology is the coarsest topology on a space X which makes a given collection of maps from X to topological spaces Xi continuous. The final topology is the finest topology on a space X which makes a given collection of maps from topological spaces Xi to X continuous.Two of the simplest examples are the natural topologies of subspaces and quotient spaces. The natural topology on a subset of a topological space is the subspace topology. This is the coarsest topology which makes the inclusion map continuous. The natural topology on a quotient of a topological space is the quotient topology. This is the finest topology which makes the quotient map continuous.Other examples include the topology induced by the Helly metric.".
- Natural_topology wikiPageID "11934923".
- Natural_topology wikiPageLength "2692".
- Natural_topology wikiPageOutDegree "22".
- Natural_topology wikiPageRevisionID "613452229".
- Natural_topology wikiPageWikiLink Category:Mathematical_structures.
- Natural_topology wikiPageWikiLink Category:Topology.
- Natural_topology wikiPageWikiLink Coarsest_topology.
- Natural_topology wikiPageWikiLink Comparison_of_topologies.
- Natural_topology wikiPageWikiLink Continuous_function.
- Natural_topology wikiPageWikiLink Continuous_function_(topology).
- Natural_topology wikiPageWikiLink Final_topology.
- Natural_topology wikiPageWikiLink Finest_topology.
- Natural_topology wikiPageWikiLink Helly_metric.
- Natural_topology wikiPageWikiLink Inclusion_map.
- Natural_topology wikiPageWikiLink Induced_topology.
- Natural_topology wikiPageWikiLink Initial_topology.
- Natural_topology wikiPageWikiLink List_of_mathematical_jargon.
- Natural_topology wikiPageWikiLink Mathematical_jargon.
- Natural_topology wikiPageWikiLink Mathematics.
- Natural_topology wikiPageWikiLink Order_topology.
- Natural_topology wikiPageWikiLink Quotient_map.
- Natural_topology wikiPageWikiLink Quotient_space_(topology).
- Natural_topology wikiPageWikiLink Quotient_topology.
- Natural_topology wikiPageWikiLink Subset.
- Natural_topology wikiPageWikiLink Subspace_topology.
- Natural_topology wikiPageWikiLink Topology.
- Natural_topology wikiPageWikiLink Total_order.
- Natural_topology wikiPageWikiLinkText "Natural topology".
- Natural_topology wikiPageWikiLinkText "natural topology".
- Natural_topology hasPhotoCollection Natural_topology.
- Natural_topology wikiPageUsesTemplate Template:Cite_book.
- Natural_topology wikiPageUsesTemplate Template:Topology-stub.
- Natural_topology subject Category:Mathematical_structures.
- Natural_topology subject Category:Topology.
- Natural_topology hypernym Topology.
- Natural_topology type Article.
- Natural_topology type Article.
- Natural_topology type Concept.
- Natural_topology type Field.
- Natural_topology comment "In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that the topology in question arises naturally or canonically (see mathematical jargon) in the given context.Note that in some cases multiple topologies seem "natural".".
- Natural_topology label "Natural topology".
- Natural_topology sameAs طوبولوجيا_طبيعية.
- Natural_topology sameAs m.02rypmd.
- Natural_topology sameAs Q6980793.
- Natural_topology sameAs Q6980793.
- Natural_topology wasDerivedFrom Natural_topology?oldid=613452229.
- Natural_topology isPrimaryTopicOf Natural_topology.