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<center>For lessons on the [[topic]] of '''''Connectedness''''', follow [https://nordan.daynal.org/wiki/index.php?title=Category:Connectedness '''''this link'''''].</center>

<center>For lessons on the [[topic]] of '''''Connectedness''''', follow [https://nordan.daynal.org/wiki/index.php?title=Category:Connectedness '''''this link'''''].</center>

==Connectedness in topology==

==Connectedness in topology==

A [http://en.wikipedia.org/wiki/Topological_space topological space] is said to be connected if it is not the union of two disjoint nonempty open sets. A set 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.

A [https://en.wikipedia.org/wiki/Topological_space topological space] is said to be connected if it is not the union of two disjoint nonempty open sets. A set 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==

==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 [http://en.wikipedia.org/wiki/Topological_space topological space], it is a connected [[space]]. Thus, manifolds, [http://en.wikipedia.org/wiki/Lie_group Lie groups], and [http://en.wikipedia.org/wiki/Graph_(mathematics) graphs] are all called connected if they are connected as [http://en.wikipedia.org/wiki/Graph_(mathematics) 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 [http://en.wikipedia.org/wiki/Vertex_(graph_theory) vertices] in the graph is joined by a path. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the [[context]] of [http://en.wikipedia.org/wiki/Graph_theory graph theory]. Graph theory also offers a [[context]]-free [[measure]] of connectedness, called the [http://en.wikipedia.org/wiki/Clustering_coefficient clustering coefficient].

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 [https://en.wikipedia.org/wiki/Topological_space topological space], it is a connected [[space]]. Thus, manifolds, [https://en.wikipedia.org/wiki/Lie_group Lie groups], and [https://en.wikipedia.org/wiki/Graph_(mathematics) graphs] are all called connected if they are connected as [https://en.wikipedia.org/wiki/Graph_(mathematics) 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 [https://en.wikipedia.org/wiki/Vertex_(graph_theory) vertices] in the graph is joined by a path. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the [[context]] of [https://en.wikipedia.org/wiki/Graph_theory graph theory]. Graph theory also offers a [[context]]-free [[measure]] of connectedness, called the [https://en.wikipedia.org/wiki/Clustering_coefficient clustering coefficient].

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 [http://en.wikipedia.org/wiki/Category_theory category theory], a category is said to be connected if each pair of objects in it is joined by a sequence of [http://en.wikipedia.org/wiki/Morphism morphism]. Thus, a category is connected if it is, [[intuitively]], all one piece.

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 [https://en.wikipedia.org/wiki/Category_theory category theory], a category is said to be connected if each pair of objects in it is joined by a sequence of [https://en.wikipedia.org/wiki/Morphism morphism]. Thus, a category is connected if it is, [[intuitively]], all one piece.

There may be [[different]] [[Thoughts|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 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.

There may be [[different]] [[Thoughts|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 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 [http://en.wikipedia.org/wiki/Path_(topology) path]-connected topological space is simply connected 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 [[sphere]] and a [http://en.wikipedia.org/wiki/Disk_(mathematics) disk] are each simply connected, while a [http://en.wikipedia.org/wiki/Torus torus] is not. As another example, a [http://en.wikipedia.org/wiki/Directed_graph directed graph] is strongly connected if each ordered pair of vertices is joined by a directed path (that is, one that "follows the arrows").

Terms involving connected are also used for properties that are related to, but clearly different from, connectedness. For example, a [https://en.wikipedia.org/wiki/Path_(topology) path]-connected topological space is simply connected 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 [[sphere]] and a [https://en.wikipedia.org/wiki/Disk_(mathematics) disk] are each simply connected, while a [https://en.wikipedia.org/wiki/Torus torus] is not. As another example, a [https://en.wikipedia.org/wiki/Directed_graph directed graph] is strongly connected if each ordered pair 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 [http://en.wikipedia.org/wiki/Totally_disconnected totally disconnected] if each of its components is a single point.

Other [[concepts]] [[express]] the way in which an object is not connected. For example, a topological space is [https://en.wikipedia.org/wiki/Totally_disconnected totally disconnected] if each of its components is a single point.

==Connectivity==

==Connectivity==

Properties and parameters based on the idea of connectedness often involve the [[word]] connectivity. For example, in graph theory, 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.

Properties and parameters based on the idea of connectedness often involve the [[word]] connectivity. For example, in graph theory, 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.

==See Also==

==See Also==

*'''''[[Interdependence]]'''''

*'''''[[Interdependence]]'''''

*[http://en.wikipedia.org/wiki/Six_degrees_of_separation '''''Six Degrees of Separation''''']

*[https://en.wikipedia.org/wiki/Six_degrees_of_separation '''''Six Degrees of Separation''''']

[[Category: Mathematics]]

[[Category: Mathematics]]