Dictionary Definition
connectedness
Noun
1 the state of being connected; "the connection
between church and state is inescapable" [syn: connection, link] [ant: disjunction]
2 a relation between things or events (as in the
case of one causing the other or sharing features with it); "there
was a connection between eating that pickle and having that
nightmare" [syn: connection, connexion] [ant: unconnectedness]
User Contributed Dictionary
Extensive Definition
this mathematics In mathematics, 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
space is said to be connected
if it cannot be contained in 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
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, manifolds, Lie groups, and
graphs
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.
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. Graph theory also offers a context-free measure of
connectedness, called the 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 category
theory, a category
is said to be connected if each pair of objects in it is joined by
a morphism. 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 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
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 disk
are each simply connected, while a torus is not. As another example,
a 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 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 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.While terminology varies, noun 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:
connectedness in German: Vernetzung
connectedness in Georgian: ბმულობა
connectedness in Romanian:
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