Connected space

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Connected space

In topology

Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
and related branches of mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a connected space is a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
which cannot be represented as the disjoint union

Disjoint union (topology)

In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology....
of two or more nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces.

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Encyclopedia

In topology

Topology

Mathematics

Topological space

Disjoint union (topology)

Path (topology)

In mathematics, a path in a topological space X is a continuous f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f....
.

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace

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 natural topology induced from that of X called the subspace topology ....
of X.

It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangle

Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
s, each of which is a space and not adjoined to the other. The space is not connected since two rectangles are disjoint. Another good example is a plane with a ring-shaped piece removed. The space is not connected since you cannot connect two points, one inside the ring and the other outside; hence the term "connect".

Formal definition

A topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
X is said to be disconnected if it is the union

Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
of two disjoint

Disjoint

Disjoint may refer to:*Disjoint sets*Disjoint union...
nonempty open set

Open set

In metric 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....
s. Otherwise, X is said to be connected. A subset

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
of a topological space is said to be connected if it is connected under its subspace topology. Some authors specifically exclude the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. 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....
with its unique topology as a connected space, but this encyclopedia does not follow that practice.

For a topological space X the following conditions are equivalent:

X is connected.
X cannot be divided into two disjoint nonempty closed set
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
s.
The only subsets of X which are both open and closed (clopen set
Clopen set

In topology, a clopen set in a topological space is a set which is both open set and closed set....
s) are X and the empty set.
The only subsets of X with empty boundary
Boundary (topology)

In topology, 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....
are X and the empty set.
X cannot be written as the union of two nonempty separated sets
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....
.
The only continuous functions from X to are constant.

The maximal

Maximal element

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S....
connected subsets of any topological space are called the connected components of the space. The components form a partition

Partition of a set

In mathematics, a partition of a Set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that cover all of X....
of the space (that is, they are disjoint

Disjoint

Disjoint may refer to:*Disjoint sets*Disjoint union...
and their union is the whole space). Every component is a closed subset of the original space. The components in general need not be open: the components of the rational number

Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s, for instance, are the one-point sets. A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X is called totally separated if, for any two elements x and y of X, there exist disjoint open neighborhoods U of x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff

Hausdorff space

Examples

The closed interval [0, 2] in the standard
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
subspace topology
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 natural topology induced from that of X called the subspace topology ....
is connected; although it can, for example, be written as the union of [0, 1) and [1, 2], the second set is not open in the aforementioned topology of [0, 2].
The union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the standard topological space [0, 1) ? (1, 2].
(0, 1) ? is disconnected.
A convex set
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
is connected; it is actually simply connected.
A Euclidean plane
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
A Euclidean plane with a straight-line removed is not connected since it consists of two half-planes.
The space of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s with the usual topology is connected.
Any topological vector space
Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topology with the algebraic concept of a vector space....
over a connected field is connected.
Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected
Totally disconnected space

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

In mathematics, the Cantor set, introduced by Germany mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties....
is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
If a space X is homotopic
Homotopy

In topology, two continuous function Function from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions....
to a connected space, then X is itself connected.
The topologist's sine curve
Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
is an example of a set that is connected but is neither path connected nor locally connected.

Path connectedness

A path
Path (topology)

In mathematics, a path in a topological space X is a continuous f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f....
from a point x to a point y in a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
X is a continuous function

Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
f from the unit interval

Unit interval

In mathematics, the unit interval is the interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1....
[0,1] to X with f(0) = x and f(1) = y. A path-component of X is an equivalence class

Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
of X under the equivalence relation

Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
defined by x is equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwise connected or 0-connected) if there is only one path-component, i.e. if there is a path joining any two points in X.

Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line

Long line (topology)

In topology, the long line is a topological space analogous to the real line, but much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology....
L* and the topologist's sine curve
Topologist's sine curve

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

However, subsets of the real line

Real line

In mathematics, the real line is simply the set R of singleton 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....
R are connected if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
they are path-connected; these subsets are the intervals

Interval (mathematics)

In mathematics, a 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....
of R. Also, open subsets of Rⁿ or Cⁿ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.

A space X is said to be arc-connected or arcwise connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism

Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
between the unit interval [0, 1] and its image f([0, 1]). It can be shown any Hausdorff space

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 neighbourhood ....
which is path-connected is also arc-connected. An example of a space which is path-connected but not arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0, 8). One endows this set with a partial order

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set ....
by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable. One then endows this set with the order topology, that is one takes the open intervals (a, b) = and the half-open intervals [0, a) = , [0', a) = as a base

Base (topology)

T1 space

In topology and related branches of mathematics, T1 spaces and R0 spaces are particular kinds of topological spaces....
space but not a Hausdorff space

Hausdorff space

Local connectedness

A topological space is said to be locally connected
Locally connected space

In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets....
at a point x if every neighbourhood of x contains a connected open neighbourhood. It is locally connected if it has a base

Base (topology)

In mathematics, a base B for a topological space X with topological space 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....
of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve

Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....
is an example of a connected space that is not locally connected.

Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rⁿ and Cⁿ, each of which is locally path-connected. More generally, any topological manifold

Topological manifold

In mathematics, a topological manifold is a Hausdorff space topological space which looks locally like Euclidean space in a sense defined below....
is locally path-connected.

Theorems

Main theorem: Let X and Y be topological spaces and let f : X ? Y be a continuous function. If X is (path-)connected then the image
Image (mathematics)

In mathematics, the image of a set under a given function is the set of all possible function outputs when taking each element of the set, successively, as the function's argument....
f(X) is (path-)connected. This result can be considered a generalization of the intermediate value theorem
Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the of a continuous function there is a corresponding value in its domain mapping to the original....
.
If is a family of connected subsets of a topological space X indexed by an arbitrary set such that for all , in , is nonempty, then is also connected.
If is a nonempty family of connected subsets of a topological space X such that is nonempty, then is also connected.
Every path-connected space is connected
Connected space/Proofs

Every path-connected space is connectedLet S be path-connected and suppose, for contradiction, that S is not connected. Then for nonempty disjoint Open sets A and B....
.
Every locally path-connected space is locally connected.
A locally path-connected space is path-connected if and only if it is connected
Connected space/Proofs

Every path-connected space is connectedLet S be path-connected and suppose, for contradiction, that S is not connected. Then for nonempty disjoint Open sets A and B....
.
The closure
Closure (topology)

In mathematics, the closure of a set S consists of all Topology glossary#Ps which are intuitively "close to S". A point which is in the closure of S is a adherent point of S....
of a connected subset is connected.
The connected components are always closed
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
(but in general not open)
The connected components of a locally connected space are also open.
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
Every quotient
Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space....
of a connected (resp. path-connected) space is connected (resp. path-connected).
Every product
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....
of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
Every manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
is locally path-connected.

Graphs

Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any n-cycle with n>3 odd) is one such example.

As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets . Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.

However, it is possible to embed a graph into such that the edges are homeomorphic to copies of the closed unit interval [0,1]. Then one can show that the graph is connected (in the graph theoretical sense) if and only if this subset of is connected w.r.t. the subspace topology.

Connected space

Formal definition

Examples

Path connectedness

Local connectedness

Theorems

Graphs

See also