Arithmetic | Topological spaces
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. (Wikipedia).
General formula for Arithmetic Progressions
Support me on Patreon: https://www.patreon.com/mathsaurus This video describes the general formulae used to describe arithmetic progressions. Visit http://www.mathsaurus.com/ for more free GCSE and A-level maths videos and resources Visit the Mathsaurus Amazon shop at https://www.amazo
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Support me on Patreon: https://www.patreon.com/mathsaurus Introduction to Arithmetic progressions, including nth term and term to term definitions. Visit http://www.mathsaurus.com/ for more free GCSE and A-level maths videos and resources. Visit the Mathsaurus Amazon shop at https://www.
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