Divisor topology

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In mathematics, more specifically general topology, the divisor topology is an example of a topology given to the set X of positive integers that are greater than or equal to two, i.e., X = {2, 3, 4, 5, …}. The divisor topology is the poset topology for the partial order relation of divisibility on the positive integers.

To give the set X a topology means to say which subsets of X 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. The set X and the empty set ∅ are open sets.

Construction[edit]

The set X and the empty set ∅ are required to be open sets, and so we define X and ∅ to be open sets in this topology. Denote by Z+ the set of positive integers, i.e., the set of positive whole number greater than or equal to one. Read the notation x|n as "x divides n", and consider the sets

S_n = \{x \in \bold{Z}^+ : x|n \}

Then the set Sn is the set of divisors of n. For different values of n, the sets Sn are used as a basis for the divisor topology.[1]

The open sets in this topology are the lower sets for the partial order defined by xy if x | y.

Properties[edit]

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