Divisor topology
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]}
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- The set X and the empty set ∅ are open sets.
Contents
Construction
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
Then the set S_{n} is the set of divisors of n. For different values of n, the sets S_{n} 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 x ≤ y if x | y.
Properties
- The set of prime numbers is dense in X. In fact, every dense open set must include every prime, and therefore X is a Baire space.^{[1]}
- X is a Kolmogorov space that is not T1. In particular, it is non-Hausdorff.
- X is second-countable.
- X is connected and locally connected.
- X is not compact, since the basic open sets S_{n} comprise an infinite covering with no finite subcovering. X is not locally compact.
- The closure of a point in x is the set of all multiples of x.
See also
- Zariski topology: A topology on the integers whose open sets are the complements of prime ideals.
References
- ^ ^{a} ^{b} ^{c} Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X