Get it on Google Play
New! Download Unionpedia on your Android™ device!
Faster access than browser!

Discrete space

+ Save concept

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. [1]

68 relations: American Mathematical Society, Axiom of choice, Base (topology), Boolean prime ideal theorem, Cantor set, Cantor space, Cardinality, Category theory, Closed set, Cofree coalgebra, Compact space, Comparison of topologies, Complete metric space, Continued fraction, Continuous function, Countable set, Counterexamples in Topology, Cylinder set, Delone set, Discrete group, Empty set, Finite set, First-countable space, Foundations of mathematics, Free object, Group (mathematics), Hausdorff space, Homeomorphism, If and only if, Integer, Irrational number, Isolated point, Lebesgue covering dimension, Lie group, Limit point, Lipschitz continuity, Locally constant function, Manifold, Mathematical structure, Meagre set, Metric map, Metric space, Morphism, Natural number, Neighbourhood (mathematics), Open set, Pontryagin duality, Product topology, Real line, Second-countable space, ..., Separation axiom, Set (mathematics), Singleton (mathematics), Springer Science+Business Media, Subset, Subspace topology, Taxicab geometry, Ternary numeral system, Topological group, Topological space, Topology, Totally bounded space, Totally disconnected space, Trivial topology, Uniform continuity, Uniform isomorphism, Uniform space, 2. Expand index (18 more) »

American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

New!!: Discrete space and American Mathematical Society · See more »

Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

New!!: Discrete space and Axiom of choice · See more »

Base (topology)

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.We are using a convention that the union of empty collection of sets is the empty set.

New!!: Discrete space and Base (topology) · See more »

Boolean prime ideal theorem

In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra.

New!!: Discrete space and Boolean prime ideal theorem · See more »

Cantor set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

New!!: Discrete space and Cantor set · See more »

Cantor space

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set.

New!!: Discrete space and Cantor space · See more »


In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

New!!: Discrete space and Cardinality · See more »

Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

New!!: Discrete space and Category theory · See more »

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

New!!: Discrete space and Closed set · See more »

Cofree coalgebra

In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space.

New!!: Discrete space and Cofree coalgebra · See more »

Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

New!!: Discrete space and Compact space · See more »

Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.

New!!: Discrete space and Comparison of topologies · See more »

Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

New!!: Discrete space and Complete metric space · See more »

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

New!!: Discrete space and Continued fraction · See more »

Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

New!!: Discrete space and Continuous function · See more »

Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

New!!: Discrete space and Countable set · See more »

Counterexamples in Topology

Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.

New!!: Discrete space and Counterexamples in Topology · See more »

Cylinder set

In mathematics, a cylinder set is the natural set in a product space.

New!!: Discrete space and Cylinder set · See more »

Delone set

In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are.

New!!: Discrete space and Delone set · See more »

Discrete group

In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology.

New!!: Discrete space and Discrete group · See more »

Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

New!!: Discrete space and Empty set · See more »

Finite set

In mathematics, a finite set is a set that has a finite number of elements.

New!!: Discrete space and Finite set · See more »

First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".

New!!: Discrete space and First-countable space · See more »

Foundations of mathematics

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

New!!: Discrete space and Foundations of mathematics · See more »

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.

New!!: Discrete space and Free object · See more »

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

New!!: Discrete space and Group (mathematics) · See more »

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.

New!!: Discrete space and Hausdorff space · See more »


In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

New!!: Discrete space and Homeomorphism · See more »

If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

New!!: Discrete space and If and only if · See more »


An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

New!!: Discrete space and Integer · See more »

Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

New!!: Discrete space and Irrational number · See more »

Isolated point

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).

New!!: Discrete space and Isolated point · See more »

Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

New!!: Discrete space and Lebesgue covering dimension · See more »

Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

New!!: Discrete space and Lie group · See more »

Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

New!!: Discrete space and Limit point · See more »

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

New!!: Discrete space and Lipschitz continuity · See more »

Locally constant function

In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant.

New!!: Discrete space and Locally constant function · See more »


In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

New!!: Discrete space and Manifold · See more »

Mathematical structure

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

New!!: Discrete space and Mathematical structure · See more »

Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sense small or negligible.

New!!: Discrete space and Meagre set · See more »

Metric map

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).

New!!: Discrete space and Metric map · See more »

Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

New!!: Discrete space and Metric space · See more »


In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.

New!!: Discrete space and Morphism · See more »

Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

New!!: Discrete space and Natural number · See more »

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

New!!: Discrete space and Neighbourhood (mathematics) · See more »

Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

New!!: Discrete space and Open set · See more »

Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.

New!!: Discrete space and Pontryagin duality · See more »

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

New!!: Discrete space and Product topology · See more »

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

New!!: Discrete space and Real line · See more »

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.

New!!: Discrete space and Second-countable space · See more »

Separation axiom

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider.

New!!: Discrete space and Separation axiom · See more »

Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

New!!: Discrete space and Set (mathematics) · See more »

Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

New!!: Discrete space and Singleton (mathematics) · See more »

Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

New!!: Discrete space and Springer Science+Business Media · See more »


In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

New!!: Discrete space and Subset · See more »

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

New!!: Discrete space and Subspace topology · See more »

Taxicab geometry

A taxicab geometry is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.

New!!: Discrete space and Taxicab geometry · See more »

Ternary numeral system

The ternary numeral system (also called base 3) has three as its base.

New!!: Discrete space and Ternary numeral system · See more »

Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

New!!: Discrete space and Topological group · See more »

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

New!!: Discrete space and Topological space · See more »


In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

New!!: Discrete space and Topology · See more »

Totally bounded space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the given context).

New!!: Discrete space and Totally bounded space · See more »

Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets.

New!!: Discrete space and Totally disconnected space · See more »

Trivial topology

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.

New!!: Discrete space and Trivial topology · See more »

Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.

New!!: Discrete space and Uniform continuity · See more »

Uniform isomorphism

In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties.

New!!: Discrete space and Uniform isomorphism · See more »

Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure.

New!!: Discrete space and Uniform space · See more »


2 (two) is a number, numeral, and glyph.

New!!: Discrete space and 2 · See more »

Redirects here:

Discrete metric, Discrete topological space, Discrete topology, Discrete uniform space, Topological discrete space.


[1] https://en.wikipedia.org/wiki/Discrete_space

Hey! We are on Facebook now! »