In 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...

, a branch of mathematics, the evenly spaced integer topology is the 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...

 on the set of 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 } generated by the family of all arithmetic progression

Arithmetic progression

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...

s. This particular topological space was introduced by where it was used to prove the infinitude of primes

Furstenberg's proof of the infinitude of primes

In number theory, Hillel Fürstenberg's proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences....

.

Construction

An arithmetic progression associated to two whole numbers a and k, with is the set of integers
To give the set Z 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 Z are open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 in a manner that satisfies 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:

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
   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.

The family of all arithmetic progressions does not satisfy these axioms: the union of arithmetic progressions need not be an arithmetic progression itself, e.g., } is not an arithmetic progression. So the evenly spaced integer topology is defined to be the topology generated by the family of arithmetic progressions. This is the coarsest topology that includes as open subsets the family of all arithmetic progressions: that is, arithmetic progressions are a subbase

Subbase

In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...

 for the topology. Since the intersection of any finite collection of arithmetic progressions is again an arithmetic progression, the family of arithmetic progressions is a base

Base (topology)

In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...

for the topology, meaning that every open set is a union of arithmetic progressions.