In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set Z+ = {1, 2, 3, 4, …}.[1] To give the set Z+ a topology means to say which subsets of Z+ are "open", and to do so in a way that the following axioms are met:[1]
The union of open sets is an open set.
The finite intersection of open sets is an open set.
Z+ and the empty set ∅ are open sets.
Construction
Given two positive integers a, b ∈ Z+, define the following congruence class:
\( U_a(b)=\{b+na \in \mathbf{Z}^+\, |\, n \in \mathbf{Z} \} \)
Then the relatively prime integer topology is the topology generated from the basis
\( \mathfrak{B} = \{U_a(b)\, |\, a,b \in \mathbf{Z}^+, (a,b)=1\} \)
and the prime integer topology is the sub-topology generated from the sub-basis
\( {\displaystyle {\mathfrak {P}}=\{U_{p}(b)\,|\,p,b\in \mathbf {Z} ^{+},p{\text{ is prime}},(p,b)=1\}} \)
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
Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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