Connectedness: Encyclopedia II - Connectedness - Other notions of connectedness

Connectedness - Other notions of connectedness

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.

Adapted from the Wikipedia article "Other notions of connectedness", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki