In mathematics, more specifically general topology

General topology

In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

, the divisor topology is an example of a topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 given to the set X of positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s that are greater than or equal to two, i.e., }. The divisor topology is the poset topology

Poset topology

In mathematics, the poset topology associated with a partially ordered set S is the Alexandrov topology on the poset of finite chains of S, ordered by inclusion.Let V be a set of vertices...

 for the partial order relation of divisibility on the positive integers.

To give the set X a topology means to say which subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s of X are "open", and to do so in a way that the following axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s are met:

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
   Empty set
   In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
   
    ∅ are open sets.

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 integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, 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 divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s of n. For different values of n, the sets S_n are used as a basis for the divisor topology.

The open sets in this topology are the lower sets for the partial order defined by

Properties

• The set of prime number
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  s is dense in X. In fact, every dense open set must include every prime, and therefore X is a Baire space
  Baire space
  In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
  
  .
• X is a Kolmogorov space
  Kolmogorov space
  In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
  
   that is not T1
  T1 space
  In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
  
  . In particular, it is non-Hausdorff
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
  .
• X is second countable.
• X is connected
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
   and locally connected.
• X is not compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
  , 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
  Zariski topology
  In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
  
  : A topology on the integers whose open sets are the complements of prime ideal
  Prime ideal
  In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
  
  s.