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Partition topology — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Partition_topology
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T14:32:00.340Z
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Partition topology — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Partition_topology (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_690b85fdd2ca (Metraipolis archive dc_690b85fdd2ca, sha256 5a57979b74b711c726317f21cf4dbfbba5bcc4e05b3285b44045834a45dfbb84)

BibTeX

@misc{dc690b85fdd2ca,
  title = {Partition topology — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Partition_topology},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_690b85fdd2ca; sha256 5a57979b74b711c726317f21cf4dbfbba5bcc4e05b3285b44045834a45dfbb84},
  year = {2026}
}

Archived text

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

In mathematics, a partition topology is a topology that can be induced on any set X {\displaystyle X} by partitioning X {\displaystyle X} into disjoint subsets P ; {\displaystyle P;} these subsets form the basis for the topology. There are two important examples which have their own names: The odd–even topology is the topology where X = N {\displaystyle X=\mathbb {N} } and P = { { 2 k − 1 , 2 k } : k ∈ N } . {\displaystyle P={\left\{~\{2k-1,2k\}:k\in \mathbb {N} \right\}}.} Equivalently, P = { { 1 , 2 } , { 3 , 4 } , { 5 , 6 } , … } . {\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.} The deleted integer topology is defined by letting X = ⋃ n ∈ N ( n − 1 , n ) ⊆ R {\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}} and P = { ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) , … } . {\displaystyle P={\left\{(0,1),(1,2),(2,3),\ldots \right\}}.} The trivial partitions yield the discrete topology (each point of X {\displaystyle X} is a set in P , {\displaystyle P,} so P = { { x } : x ∈ X } {\displaystyle P=\{~\{x\}~:~x\in X~\}} ) or indiscrete topology (the entire set X {\displaystyle X} is in P , {\displaystyle P,} so P = { X } {\displaystyle P=\{X\}} ). Any set X {\displaystyle X} with a partition topology generated by a partition P {\displaystyle P} can be viewed as a pseudometric space with a pseudometric given by: d ( x , y ) = { 0 if x and y are in the same partition element 1 otherwise . {\displaystyle d(x,y)={\begin{cases}0&{\text{if }}x{\text{ and }}y{\text{ are in the same partition element}}\\1&{\text{otherwise}}.\end{cases}}} This is not a metric unless P {\displaystyle P} yields the discrete topology. The partition topology provides an important example of the independence of various separation axioms. Unless P {\displaystyle P} is trivial, at least one set in P {\displaystyle P} contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X {\displaystyle X} is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, X {\displaystyle X} is regular, completely regular, normal and completely normal. X / P {\displaystyle X/P} is the discrete topology.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «topology»).
Fonte: Wikipedia in inglese, voce «Partition topology» — https://en.wikipedia.org/wiki/Partition_topology
Wikidata Q17127067 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q17127067
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Partition_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 1301446555, consultata il 2026-10-07T14:08:57.805Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 69d24e97924eb8aea3b026b8cd05d882e5c0fbac8788dbe32421a09d66d4b62f