In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
,
connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is
connected; otherwise it is
disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a
component (or
connected component).
A
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be
connectedIn topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
if it is not the union of two disjoint nonempty
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
s.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
,
connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is
connected; otherwise it is
disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a
component (or
connected component).
Connectedness in topology
A
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be
connectedIn topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
if it is not the union of two disjoint nonempty
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
s. A set is open if it contains no point lying on its
boundaryIn 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...
; thus, in an informal, intuitive sense, the fact that a space can be partitioned into disjoint open sets suggests that the boundary between the two sets is not part of the space, and thus splits it into two separate pieces.
Other notions of connectedness
Fields of mathematics are typically concerned with special kinds of objects. Often such an object is said to be
connected if, when it is considered as a topological space, it is a connected space. Thus,
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s,
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and
graphsIn 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...
are all called
connected if they are connected as topological spaces, and their components are the topological components. Sometimes it is convenient to restate the definition of connectedness in such fields. For example, a graph is said to be
connected if each pair of
verticesIn graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
in the graph is joined by a
pathIn graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of
graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
. Graph theory also offers a context-free measure of connectedness, called the
clustering coefficientA fundamental measure that has long received attention in both theoretical and empirical network research is the clustering coefficient. This measure assesses the degree to which nodes tend to cluster together...
.
Other fields of mathematics are concerned with objects that are rarely considered as topological spaces. Nonetheless, definitions of
connectedness often reflect the topological meaning in some way. For example, in
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, a
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
is said to be
connectedIn category theory, a branch of mathematics, a connected category is a category in which, for every two objects X and Y there is a finite sequence of objectswith morphismsor...
if each pair of objects in it is joined by a sequence of
morphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...
. Thus, a category is connected if it is, intuitively, all one piece.
There may be different notions of
connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space
connected if each pair of points in it is joined by a
pathIn mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; spaces with this property are said to be
path connected.
Terms involving
connected are also used for properties that are related to, but clearly different from, connectedness. For example, a path-connected topological space is
simply connectedIn topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other while preserving the two endpoints in question....
if each loop (path from a point to itself) in it is contractible; that is, intuitively, if there is essentially only one way to get from any point to any other point. Thus, a
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
and a
diskIn geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
are each simply connected, while a
torusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...
is not. As another example, a
directed graphA directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,* a set A of ordered pairs of vertices, called arcs, directed edges, or arrows .It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of...
is
strongly connectedA directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex. In particular, this means paths in each direction; a path from a to b and also a path from b to a....
if each
ordered pairIn mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate of the pair as well as the second coordinate...
of vertices is joined by a directed path (that is, one that "follows the arrows").
Other concepts express the way in which an object is
not connected. For example, a topological space is
totally disconnected if each of its components is a single point.
Connectivity
Properties and parameters based on the idea of connectedness often involve the word
connectivity. For example, in
graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, a connected graph is one from which we must remove at least one vertex to create a disconnected graph. In recognition of this, such graphs are also said to be
1-connected. Similarly, a graph is
2-connected if we must remove at least two vertices from it, to create a disconnected graph. A
3-connected graph requires the removal of at least three vertices, and so on. The
connectivityIn mathematics and computer science, connectivity is one of the basic concepts of graph theory. It is closely related to the theory of network flow problems...
of a graph is the minimum number of vertices that must be removed, to disconnect it. Equivalently, the connectivity of a graph is the greatest integer
k for which the graph is
k-connected.
While terminology varies,
nounIn linguistics, a noun is a member of a large, open lexical category whose members can occur as the main word in the subject of a clause, the object of a verb, or the object of a preposition....
forms of connectedness-related properties often include the term
connectivity. Thus, when discussing simply connected topological spaces, it is far more common to speak of
simple connectivity than
simple connectedness. On the other hand, in fields without a formally defined notion of
connectivity, the word may be used as a synonym for
connectedness.
Another example of connectivity can be found in regular tilings. Here, the connectivity describes the number of neighbors accessible from a single
tileA tile is a manufactured piece of hard-wearing material such as ceramic, stone, metal, or even glass. Tiles are generally used for covering roofs, floors, and walls, showers, or other objects such as tabletops...
:
See also
- connected category
In category theory, a branch of mathematics, a connected category is a category in which, for every two objects X and Y there is a finite sequence of objectswith morphismsor...
- connected component (graph theory)
In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and to which no more vertices or edges can be added while preserving its connectivity. That is, it is a maximal connected subgraph. For example, the graph...
- connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...
- cross-link
Cross-links are bonds that link one polymer chain to another. They can be covalent bonds or ionic bonds. "Polymer chains" can refer to synthetic polymers or natural polymers . When the term "cross-linking" is used in the synthetic polymer science field, it usually refers to the use of...
- network
In graph theory, a network is a digraph with weighted edges. These networks have become an especially useful concept in analysing the interaction between biology and mathematics...
- Scale-free network
A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P of nodes in the network having k connections to other nodes goes for large values of k as P ~ k
−γ where γ is a constant whose value is typically in the range...
- simply connected
- small-world network
In mathematics, physics and sociology a small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps...
- strongly connected component
A directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex. In particular, this means paths in each direction; a path from a to b and also a path from b to a....
- totally disconnected