mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

is said to be '' connected'' if it is not the union of two disjoint nonempty

open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

s. A

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

is open if it contains no point lying on its 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,

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s, and

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 discret ...

s 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 vertices in the graph is joined by a

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...

. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

. Graph theory also offers a context-free measure of connectedness, called the

clustering coefficient In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups ...

. 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 theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

is said to be '' connected'' if each pair of objects in it is joined by a sequence of

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s. 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

. However this condition turns out to be stronger than 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 In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...

''. While not all connected spaces are path connected, all path connected spaces are 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 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 (topology), path between two points can be continuously transformed into any other such path while preserving ...

'' if each loop (path from a point to itself) in it is

contractible 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 t ...

; that is, intuitively, if there is essentially only one way to get from any point to any other point. Thus, a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

and a disk are each simply connected, while a

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

is not. As another example, a directed graph is ''

strongly connected In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of a directed graph form a partition into subgraphs that are thems ...

'' if each

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

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 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) ...

'' 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

, 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 '' connectivity'' 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,

noun In grammar, a noun is a word that represents a concrete or abstract thing, like living creatures, places, actions, qualities, states of existence, and ideas. A noun may serve as an Object (grammar), object or Subject (grammar), subject within a p ...

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

tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wal ...

: Image:Triangular_3_connectivity.svg, 3-connectivity in a triangular tiling, Image:Square_4_connectivity.svg, 4-connectivity in a

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

, Image:Hexagonal_connectivity.svg, 6-connectivity in a

hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a Truncation (geometry), truncated triangular tiling ...

, Image:Square_8_connectivity.svg, 8-connectivity in a

(note that distance equality is not kept)

References

{{Reflist Mathematical terminology

Connectedness in topology

Other notions of connectedness

Connectivity

See also

References