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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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 poin ...
that cannot be represented as the union of two or more disjoint non-empty 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, and n-connected. Another related notion is '' locally connected'', 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 poin ...
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 of ...
of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (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 sets) 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 (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, 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 ra ...
s are the one-point sets ( singletons), which are not open. Proof: Any two distinct rational numbers q_1 are in different components. Take an irrational number q_1 < r < q_2, and then set A = \ and B = \. Then (A,B) is a separation of \Q, and q_1 \in A, q_2 \in B. Thus each component is a one-point set. Let \Gamma_x be the connected component of x in a topological space X, and \Gamma_x' be the intersection of all clopen sets containing x (called quasi-component of x.) Then \Gamma_x \subset \Gamma'_x where the equality holds if X is compact Hausdorff or locally connected.


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, 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 standard_subspace_topology.html" ;"title="Euclidean_space.html" ;"title=", 2) 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 [0, 1) and [1, 2), the second set is not open in the chosen 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) \cup (1, 2]. * (0, 1) \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 ...
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. * \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. Every ...
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, 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, over a connected field (such as \R or \Complex), is simply connected. * Every
discrete topological 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 , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space. * On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space. * The Cantor set 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 is an example of a set that is connected but is neither path connected nor locally connected. * The general linear group \operatorname(n, \R) (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(n, \Complex) 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 algebrai ...
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 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. Various equivalent character ...
over \R has constant rank. *# \R has no idempotent \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 Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology 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 poin ...
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 ...
,1/math> to X with f(0)=x and f(1)=y. A of X is an equivalence class of X under the equivalence relation 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 L^* and the topologist's sine curve. 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 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 example ...
s.


Arc connectedness

A space X is said to be arc-connected or arcwise connected if any two topologically distinguishable points can be joined by an arc, which is an embedding f : , 1\to X. 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 In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
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 In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff ...
; 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 In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff ...
. 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 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 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 (0,1) \cup (2,3). A classical example of a connected space that is not locally connected is the so called topologist's sine curve, 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
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=(0,1) \cup (1,2). 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 unions. 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(U) \cup q(V) is a separation of the quotient space (since q(U), q(V) are disjoint and open in the quotient space). The set difference 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(X) 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 impor ...
. *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 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

Graphs 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 poin ...
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 spaces are also connected. * Since a simply connected 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. 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.


See also

* * * * * * * *


References


Further reading

* * * * . {{DEFAULTSORT:Connected Space General topology Properties of topological spaces