Prime integer topology

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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]

  1. The union of open sets is an open set.
  2. The finite intersection of open sets is an open set.
  3. Z+ and the empty set ∅ are open sets.

Construction[edit]

Given two positive integers a, bZ+, 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

\mathfrak{P} = \{U_p(b)\, |\, p,b \in \mathbf{Z}^+, p \text{ is prime}\}

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[edit]

References[edit]

  1. ^ a b c Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X