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Arithmetic progression topologies — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Arithmetic_progression_topologies
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T11:46:17.606Z
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Arithmetic progression topologies — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Arithmetic_progression_topologies (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_bedded9de575 (Metraipolis archive dc_bedded9de575, sha256 235642b891c9dcbefba48965a319b3698fd46e2daf8d8876b3bc071f315896d6)

BibTeX

@misc{dcbedded9de575,
  title = {Arithmetic progression topologies — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Arithmetic_progression_topologies},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_bedded9de575; sha256 235642b891c9dcbefba48965a319b3698fd46e2daf8d8876b3bc071f315896d6},
  year = {2026}
}

Archived text

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

In general topology and number theory, branches of mathematics, one can define various topologies on the set Z {\displaystyle \mathbb {Z} } of integers or the set Z > 0 {\displaystyle \mathbb {Z} _{>0}} of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form { b , b + a , b + 2 a , . . . } {\displaystyle \{b,b+a,b+2a,...\}} or { . . . , b − 2 a , b − a , b , b + a , b + 2 a , . . . } . {\displaystyle \{...,b-2a,b-a,b,b+a,b+2a,...\}.} The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on Z {\displaystyle \mathbb {Z} } , and the Golomb topology and the Kirch topology on Z > 0 {\displaystyle \mathbb {Z} _{>0}} . Precise definitions are given below. Hillel Furstenberg introduced the first topology in order to provide a "topological" proof of the infinitude of the set of primes. The second topology was studied by Solomon Golomb and provides an example of a countably infinite Hausdorff space that is connected. The third topology, introduced by A.M. Kirch, is an example of a countably infinite Hausdorff space that is both connected and locally connected. These topologies also have interesting separation and homogeneity properties. The notion of an arithmetic progression topology can be generalized to arbitrary Dedekind domains.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «topology»).
Fonte: Wikipedia in inglese, voce «Arithmetic progression topologies» — https://en.wikipedia.org/wiki/Arithmetic_progression_topologies
Wikidata Q3993517 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q3993517
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Arithmetic_progression_topologies?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 1309317919, consultata il 2026-10-07T11:20:21.605Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 51ff6a3e511f946a11b6af926884c67589a9c9634997f5aa544cdcdbb5ab7b72