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# 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. [1]

77 relations: Annulus (mathematics), Base (topology), Boundary (topology), Cantor set, Charles Weibel, Clopen set, Closed set, Closure (topology), Comb space, Complement (set theory), Connected component (graph theory), Connected space, Connectedness locus, Continuous function, Contractible space, Convex set, Covering space, Cycle graph, Discrete space, Discrete two-point space, Discrete valuation ring, Disjoint sets, Disk (mathematics), Empty set, Equivalence class, Equivalence relation, Euclidean space, Extremally disconnected space, Finite topological space, General linear group, Genus (mathematics), George F. Simmons, Graph (discrete mathematics), Hausdorff space, Homeomorphism, Homotopy, Hyperconnected space, Idempotence, If and only if, Image (mathematics), Intermediate value theorem, Interval (mathematics), Local ring, Locally connected space, Long line (topology), Manifold, Mathematics, Maximal and minimal elements, N-connected space, Open set, ... Expand index (27 more) »

## Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring" is anulus/annulus, with plural anuli/annuli) is a ring-shaped object, a region bounded by two concentric circles.

## 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.

## Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

## 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.

## Charles Weibel

Charles Alexander Weibel (born October 28, 1950 in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.

## Clopen set

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.

## Closed set

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

## Closure (topology)

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

## Comb space

In mathematics, particularly topology, a comb space is a subspace of \R^2 that looks rather like a comb.

## Complement (set theory)

In set theory, the complement of a set refers to elements not in.

## Connected component (graph theory)

In graph theory, a connected component (or just component) of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph.

## 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 locus

In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is connected.

## 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.

## Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.

## Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

## Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

## Cycle graph

In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain.

## 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.

## Discrete two-point space

In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space.

## Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

## Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common.

## Disk (mathematics)

In geometry, a disk (also spelled disc).

## 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.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Extremally disconnected space

In mathematics, a topological space is termed extremally disconnected if the closure of every open set in it is open.

## Finite topological space

In mathematics, a finite topological space is a topological space for which the underlying point set is finite.

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## George F. Simmons

George Finlay Simmons (born 1925) is an American mathematician who worked in topology and classical analysis.

## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

## 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.

## Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

## Hyperconnected space

In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint).

## Idempotence

Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.

## 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.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

## Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Local ring

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

## Locally connected space

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.

## Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".

## Manifold

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

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

## N-connected space

In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness (sometimes, n-simple connectedness) generalizes the concepts of path-connectedness and simple connectedness.

## Open set

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

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Path (topology)

In mathematics, a path in a topological space X is a continuous function f from the unit interval I.

## Pixel connectivity

In image processing and image recognition, pixel connectivity is the way in which pixels in 2-dimensional (or voxels in 3-dimensional) images relate to their neighbors.

## 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.

## Projective module

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules.

## Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real line

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

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Separated sets

In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching.

## Sierpiński space

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.

## Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

## Singleton (mathematics)

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

## 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).

## 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 a neighborhood not containing the other.

## Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

## Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.

## 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.

## Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

## Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.

## 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 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.

## Uniformly connected space

In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Unit interval

In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.

## References

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