Arithmetic progression topologies

In general topology and number theory, branches of mathematics, one can define various topologies on the set ${\displaystyle \mathbb {Z} }$ of integers or the set ${\displaystyle \mathbb {Z} _{>0}}$ of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form ${\displaystyle \{b,b+a,b+2a,...\}}$ or ${\displaystyle \{...,b-2a,b-a,b,b+a,b+2a,...\}.}$ The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on ${\displaystyle \mathbb {Z} }$, and the Golomb topology and the Kirch topology on ${\displaystyle \mathbb {Z} _{>0}}$. Precise definitions are given below.

Hillel Furstenberg^[1] 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^[2] and provides an example of a countably infinite Hausdorff space that is connected. The third topology, introduced by A.M. Kirch,^[3] 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.