# discrete topology

In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.

## Definitions

Given a set X:
• the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology;
• the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
• the discrete metric on X is defined by
for any . In this case is called a discrete metric space or a space of isolated points.

A metric space is said to be uniformly discrete if there exists such that, for any , one has either or . The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set {1, 1/2, 1/4, 1/8, ...} of real numbers.

## Properties

The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one another. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := {1/n : n = 1,2,3,...} (with metric inherited from the real line and given by d(x,y) = |x − y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformly discrete or metrically discrete.

Any function from a discrete topological space to another topological space is continuous, and any function from a discrete uniform space to another uniform space is uniformly continuous. That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.

With metric spaces, things are more complicated, because there are several categories of metric spaces, depending on what is chosen for the morphisms. Certainly the discrete metric space is free when the morphisms are all uniformly continuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniform or topological structure. Categories more relevant to the metric structure can be found by limiting the morphisms to Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more than one element). However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short.

Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every point in Y has a neighborhood on which f is constant.

## Uses

A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric. For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example combined with Pontryagin duality.

A 0-dimensional manifold (or differentiable or analytical manifold) is nothing but a discrete topological space. In the spirit of the previous paragraph, we can therefore view any discrete group as a 0-dimensional Lie group.

While discrete spaces are not very exciting from a topological viewpoint, one can easily construct interesting spaces from them. For instance, a product of countably infinitely many copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinitely many copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by ternary notation of numbers. (See Cantor space.)

In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, which is a weak form of choice.

## Indiscrete spaces

Main article: Trivial topology

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the least possible number of open sets (just the empty set and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.

## References

1. ^ Stanislaw Ulam's autobiography, Adventures of a Mathematician.
Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x
subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.
(See also Subset for the uncapitalized use of the word "superset" in mathematics.)

SuperSet Software was a group founded by friends and former Eyring Research Institute (ERI) co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst.
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x
In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space.
For Cauchy completion in category theory, see Karoubi envelope.

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in
In mathematics, a singleton is a set with exactly one element. For example, the set is a singleton.

## Properties

Note that a set such as is also a singleton: the only element is a set (which itself is however not a singleton).
In mathematics, informally speaking, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x as well as one pleases.
In mathematics, a base (or basis) 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.
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.
separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

The separation axioms are axioms only in the sense that, when defining the notion of topological space, you could add these conditions as extra axioms to get a more
Topological spaces in
separation axiom

Kolmogorov (T0) version
T0 | T1 | T2 | T | completely T2
T3 | T | T4 | T5 | T6
Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).
If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
In mathematics, a set is called finite if there is a bijection between the set and some set of the form where n is a natural number. (The value n = 0 is allowed; that is, the empty set is finite.) An infinite set is a set which is not finite.
For Cauchy completion in category theory, see Karoubi envelope.

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in
In topology and related branches of mathematics, a totally bounded space, or precompact space (note though that some sources such as PlanetMath take precompact to mean relatively compact), is a space that can be covered by finitely many subsets of any fixed size.
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base).