Prime integer topology
In mathematics, specifically general topology, the prime integer topology and the relatively prime integer topology are topologies on the set of positive integers, i.e. the set Z+ = {1, 2, 3, 4, ...}.[1]
Construction
Given two positive integers a and b, define the following congruence class:
- [math]\displaystyle{ U_a(b) = \{b+na \in \mathbf{Z}^+ \mid n \in \mathbf{Z}\}. }[/math]
Then the relatively prime integer topology is the topology generated from the basis
- [math]\displaystyle{ \mathfrak{B} = \{U_a(b) \mid a,b \in \mathbf{Z}^+, \,\operatorname{gcd}(a,b)=1\}, }[/math]
where [math]\displaystyle{ \operatorname{gcd} }[/math] is the greatest common divisor function, and the prime integer topology is the topology generated from the subbasis
- [math]\displaystyle{ \mathfrak{P} = \{U_p(b) \mid p,b \in \mathbf{Z}^+, \,p \text{ is prime}, \,\operatorname{gcd}(p,b)=1\}. }[/math]
The set of positive integers with the relatively prime integer topology or with the prime integer topology are examples of topological spaces that are Hausdorff but not regular.[1]
See also
- Fürstenberg's proof of the infinitude of primes
References
- ↑ 1.0 1.1 Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
it:Topologia degli interi equispaziati
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