Connectedness in topologyA
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 connectednessFields 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
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
rdered
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:
3-connectivity in a
triangular tiling,
|
4-connectivity in a
square tiling,
|
6-connectivity in a
hexagonal tiling,
|
8-connectivity in a
square tiling (note that distance equity is not kept)
|
|
