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# Discrete space

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, ... 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.

## 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.

## 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.

## Boolean prime ideal theorem

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

## 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.

## 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.

## Cardinality

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

## 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).

## Closed set

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

## 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.

## 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).

## 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.

## 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).

## 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.

## 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.

## 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.

## 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.

## Cylinder set

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

## 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.

## 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.

## 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.

## Finite set

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

## First-countable space

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

## 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.

## Free object

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

## 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.

## 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.

## Homeomorphism

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.

## 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.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## 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.

## 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).

## 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.

## 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.

## 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.

## Lipschitz continuity

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

## 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.

## Manifold

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

## 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.

## 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.

## 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).

## Metric space

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

## Morphism

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

## 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").

## Neighbourhood (mathematics)

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

## Open set

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

## 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.

## 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.

## Real line

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

## 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.

## 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.

## Set (mathematics)

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

## Singleton (mathematics)

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

## 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.

## Subset

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.

## 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).

## 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.

## Ternary numeral system

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

## 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.

## 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.

## Topology

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.

## 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).

## 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.

## 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.

## 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.

## 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.

## Uniform space

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

## 2

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

## References

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