# Connectedness

In mathematics, **connectedness**[1] 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 is not the union of two disjoint nonempty open sets[2]. 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 sequence of morphisms. 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 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*. 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* 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[3]. 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:

## See also

## References

- "the definition of connectedness".
*Dictionary.com*. Retrieved 2016-06-15. - Munkres, James (2000).
*Topology*. Pearson. p. 148. ISBN 978-0131816299. - Bondy, J.A.; Murty, U.S.R. (1976).
*Graph Theory and Applications*. New York, NY: Elsevier Science Publishing Co. p. 42. ISBN 0444194517.