In general topology and number theory, branches of mathematics, one can define various topologies on the set of integers or the set of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form or The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on , and the Golomb topology and the Kirch topology on . Precise definitions are given below. The notion of an arithmetic progression topology can be generalized to arbitrary Dedekind domains.
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| - Arithmetic progression topologies (en)
- Topologie des entiers uniformément espacés (fr)
- Topologia degli interi equispaziati (it)
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| - En topologie, une branche des mathématiques, la topologie des entiers uniformément espacés est la topologie sur l'ensemble des entiers relatifs ℤ = {…, −2, −1, 0, 1, 2, …} engendrée par la famille de toutes les progressions arithmétiques. C'est un cas spécial de topologie profinie sur un groupe. Cet espace topologique particulier a été introduit en 1955 par Furstenberg qui l'a utilisée pour démontrer l'infinité des nombres premiers. (fr)
- In topologia generale, una branca della matematica, la topologia degli interi equispaziati è la topologia sull'insieme dei numeri interi generata dalla famiglia delle progressioni aritmetiche. Questa particolare topologia è stata introdotta da Fürstenberg nel 1955 per provare l'infinità dei numeri primi. (it)
- In general topology and number theory, branches of mathematics, one can define various topologies on the set of integers or the set of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form or The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on , and the Golomb topology and the Kirch topology on . Precise definitions are given below. The notion of an arithmetic progression topology can be generalized to arbitrary Dedekind domains. (en)
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| - In general topology and number theory, branches of mathematics, one can define various topologies on the set of integers or the set of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form or The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on , and the Golomb topology and the Kirch topology on . 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. (en)
- En topologie, une branche des mathématiques, la topologie des entiers uniformément espacés est la topologie sur l'ensemble des entiers relatifs ℤ = {…, −2, −1, 0, 1, 2, …} engendrée par la famille de toutes les progressions arithmétiques. C'est un cas spécial de topologie profinie sur un groupe. Cet espace topologique particulier a été introduit en 1955 par Furstenberg qui l'a utilisée pour démontrer l'infinité des nombres premiers. (fr)
- In topologia generale, una branca della matematica, la topologia degli interi equispaziati è la topologia sull'insieme dei numeri interi generata dalla famiglia delle progressioni aritmetiche. Questa particolare topologia è stata introdotta da Fürstenberg nel 1955 per provare l'infinità dei numeri primi. (it)
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