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Natural topology — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Natural_topology
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T14:32:01.143Z
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Natural topology — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Natural_topology (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_e5bf49fc72fa (Metraipolis archive dc_e5bf49fc72fa, sha256 17e0b6ca4efa5310818cfffa8cb6435e12908cd82969144be606918eef9bef85)

BibTeX

@misc{dce5bf49fc72fa,
  title = {Natural topology — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Natural_topology},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_e5bf49fc72fa; sha256 17e0b6ca4efa5310818cfffa8cb6435e12908cd82969144be606918eef9bef85},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Natural_topology. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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. Another example is that any metric space has a natural topology induced by its metric.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «topology»).
Fonte: Wikipedia in inglese, voce «Natural topology» — https://en.wikipedia.org/wiki/Natural_topology
Wikidata Q6980793 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q6980793
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Natural_topology?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1340292142, consultata il 2026-10-07T14:08:57.809Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 113a9f9dedab40755dbfd89a554b48c6f5296f017a3bbc9df7eeb793030046c9