TheInfoList

OR:

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of mathematics, a connected space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that cannot be represented as the union of two or more disjoint
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space $X$ is a if it is a connected space when viewed as a subspace of $X$. Some related but stronger conditions are path connected,
simply connected 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 spac ...
, and $n$-connected. Another related notion is ''
locally connected 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. Background Throughout the history of topology, connectedness ...
'', which neither implies nor follows from connectedness.

# Formal definition

A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
$X$ is said to be if it is the union of two disjoint non-empty open sets. Otherwise, $X$ is said to be connected. A
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
(with its unique topology) as a connected space, but this article does not follow that practice. For a topological space $X$ the following conditions are equivalent: #$X$ is connected, that is, it cannot be divided into two disjoint non-empty open sets. #The only subsets of $X$ which are both open and closed (
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. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
s) are $X$ and the empty set. #The only subsets of $X$ with empty boundary are $X$ and the empty set. #$X$ cannot be written as the union of two non-empty
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. The notion of when two sets a ...
(sets for which each is disjoint from the other's closure). #All
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
functions from $X$ to $\$ are constant, where $\$ is the two-point space endowed with the discrete topology. Historically this modern formulation of the notion of connectedness (in terms of no partition of $X$ into two separated sets) first appeared (independently) with N.J. Lennes,
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
, and Felix Hausdorff at the beginning of the 20th century. See for details.

## Connected components

Given some point $x$ in a topological space $X,$ the union of any collection of connected subsets such that each contains $x$ will once again be a connected subset. The connected component of a point $x$ in $X$ is the union of all connected subsets of $X$ that contain $x;$ it is the unique largest (with respect to $\subseteq$) connected subset of $X$ that contains $x.$ The maximal connected subsets (ordered by inclusion $\subseteq$) of a non-empty topological space are called the connected components of the space. The components of any topological space $X$ form a partition of $X$: they are disjoint, non-empty and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s are the one-point sets ( singletons), which are not open. Proof: Any two distinct rational numbers

## Disconnected spaces

A space in which all components are one-point sets is called . Related to this property, a space $X$ is called if, for any two distinct elements $x$ and $y$ of $X$, there exist disjoint open sets $U$ containing $x$ and $V$ containing $y$ 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 In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
, 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, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

# Examples

* The closed interval in the Euclidean space">standard subspace topology">Euclidean_space.html" ;"title=", 2) in the Euclidean space">standard subspace topology is connected; although it can, for example, be written as the union of $\left[0, 1\right)$ and $\left[1, 2\right),$ the second set is not open in the chosen topology of $\left[0, 2\right).$ * The union of $\left[0, 1\right)$ and $\left(1, 2\right]$ is disconnected; both of these intervals are open in the standard topological space $\left[0, 1\right) \cup \left(1, 2\right].$ * $\left(0, 1\right) \cup \$ is disconnected. * A convex set, convex subset of $\R^n$ is connected; it is actually Simply connected set, simply connected. * A
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
excluding the origin, $\left(0, 0\right),$ 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. * $\R$, the space of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s with the usual topology, is connected. * The Sorgenfrey line is disconnected. * If even a single point is removed from $\mathbb$, the remainder is disconnected. However, if even a countable infinity of points are removed from $\R^n$, where $n \geq 2,$ the remainder is connected. If $n\geq 3$, then $\R^n$ remains simply connected after removal of countably many points. * Any
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
, e.g. any
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
or
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
, over a connected field (such as $\R$ or $\Complex$), is simply connected. * Every discrete topological space with at least two elements is disconnected, in fact such a space is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
. The simplest example is the
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. The points can be denoted by the symbols 0 and 1. Properties Any disconnected space has a continuous mapp ...
. * On the other hand, a finite set might be connected. For example, the spectrum of a
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
consists of two points and is connected. It is an example of a
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. It is the smallest example of a topological space which is neither trivial nor discrete. It is named ...
. * The
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Th ...
is totally disconnected; since the set contains uncountably many points, it has uncountably many components. * If a space $X$ is homotopy equivalent to a connected space, then $X$ is itself connected. * The
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. It can be defined as the graph of the functi ...
is an example of a set that is connected but is neither path connected nor locally connected. * The
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. This forms a group, because the product of two invertible matrices is again invertible, ...
$\operatorname\left(n, \R\right)$ (that is, the group of $n$-by-$n$ real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, $\operatorname\left(n, \Complex\right)$ is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected. * The spectra of commutative
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 nu ...
and integral domains are connected. More generally, the following are equivalent *# The spectrum of a commutative ring $\R$ is connected *# Every finitely generated projective module over $\R$ has constant rank. *# $\R$ has no
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
$\ne 0, 1$ (i.e., $\R$ is not a product of two rings in a nontrivial way). An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the
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 t ...
induced by two-dimensional Euclidean space.

# Path connectedness

A is a stronger notion of connectedness, requiring the structure of a path. A path from a point $x$ to a point $y$ in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
$X$ is a continuous function $f$ from the
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. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of $X$ under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
which makes $x$ 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 $\mathbf$-connected) if there is exactly one path-component, i.e. if there is a path joining any two points in $X$. Again, many authors exclude the empty space (by this definition, however, the empty space is not path-connected because it has zero path-components; there is a unique equivalence relation on the empty set which has zero equivalence classes). 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 or longline may refer to: *'' Long Line'', an album by Peter Wolf *Long line (topology), or Alexandroff line, a topological space * Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...
$L^*$ and the
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. It can be defined as the graph of the functi ...
. Subsets of the real line $\R$ are connected
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
they are path-connected; these subsets are the
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval est ...
of $R$. Also, open subsets of $\R^n$ or $\C^n$ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for
finite topological space In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples ...
s.

# Arc connectedness

A space $X$ is said to be arc-connected or arcwise connected if any two
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
points can be joined by an arc, which is an
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
. An arc-component of $X$ is a maximal arc-connected subset of $X$; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable. Every Hausdorff space that is path-connected is also arc-connected; more generally this is true for a $\Delta$-Hausdorff space, which is a space where each image of a path is closed. An example of a space which is path-connected but not arc-connected is given by the line with two origins; its two copies of $0$ can be connected by a path but not by an arc. Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let $X$ be the line with two origins. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces: * Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality. * Arc-components may not be disjoint. For example, $X$ has two overlapping arc-components. * Arc-connected product space may not be a product of arc-connected spaces. For example, $X \times \mathbb$ is arc-connected, but $X$ is not. * Arc-components of a product space may not be products of arc-components of the marginal spaces. For example, $X \times \mathbb$ has a single arc-component, but $X$ has two arc-components. *If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of $X$ intersect, but their union is not arc-connected.

# Local connectedness

A topological space is said to be
locally connected 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. Background Throughout the history of topology, connectedness ...
at a point $x$ if every neighbourhood of $x$ contains a connected open neighbourhood. It is locally connected if it has a base 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. Similarly, a topological space is said to be 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 $\R^n$ and $\C^n$, each of which is locally path-connected. More generally, any
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout m ...
is locally path-connected. Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two separated intervals in $\R$, such as $\left(0,1\right) \cup \left(2,3\right)$. A classical example of a connected space that is not locally connected is the so called
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. It can be defined as the graph of the functi ...
, defined as $T = \ \cup \left\$, with the
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
induced by inclusion in $\R^2$.

# Set operations

The intersection of connected sets is not necessarily connected. The union of connected sets is not necessarily connected, as can be seen by considering $X=\left(0,1\right) \cup \left(1,2\right)$. Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets $U$ and $V$. This means that, if the union $X$ is disconnected, then the collection $\$ can be partitioned to two sub-collections, such that the unions of the sub-collections are disjoint and open in $X$ (see picture). This implies that in several cases, a union of connected sets necessarily connected. In particular: # If the common intersection of all sets is not empty ($\bigcap X_i \neq \emptyset$), then obviously they cannot be partitioned to collections with
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
s. Hence the union of connected sets with non-empty intersection is connected. # If the intersection of each pair of sets is not empty ($\forall i,j: X_i \cap X_j \neq \emptyset$) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected. # If the sets can be ordered as a "linked chain", i.e. indexed by integer indices and $\forall i: X_i \cap X_ \neq \emptyset$, then again their union must be connected. # If the sets are pairwise-disjoint and the quotient space $X / \$ is connected, then must be connected. Otherwise, if $U \cup V$ is a separation of then $q\left(U\right) \cup q\left(V\right)$ is a separation of the quotient space (since $q\left(U\right), q\left(V\right)$ are disjoint and open in the quotient space). The
set difference In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
of connected sets is not necessarily connected. However, if $X \supseteq Y$ and their difference $X \setminus Y$ is disconnected (and thus can be written as a union of two open sets $X_1$ and $X_2$), then the union of $Y$ with each such component is connected (i.e. $Y \cup X_$ is connected for all $i$).

# Theorems

*Main theorem of connectedness: Let $X$ and $Y$ be topological spaces and let $f:X\rightarrow Y$ be a continuous function. If $X$ is (path-)connected then the image $f\left(X\right)$ is (path-)connected. This result can be considered a generalization of the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impo ...
. *Every path-connected space is connected. *Every locally path-connected space is locally connected. *A locally path-connected space is path-connected if and only if it is connected. *The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected. *The connected components are always
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
(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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected). *Every product 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 is locally path-connected. *Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected *Continuous image of arc-wise connected set is arc-wise connected.

# Graphs

Graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
s 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, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

# Stronger forms of connectedness

There are stronger forms of connectedness for
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, for instance: * If there exist no two disjoint non-empty open sets in a topological space $X$, $X$ must be connected, and thus
hyperconnected space In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
s are also connected. * Since a
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 ...
is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected. *Yet stronger versions of connectivity include the notion of a
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. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...
. Every contractible space is path connected and thus also connected. In general, any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above-mentioned
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. It can be defined as the graph of the functi ...
.