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Connectedness - Other notions of connectedness |  | Connectedness - Other notions of connectedness: Encyclopedia II - 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; ...
See also:Connectedness, Connectedness - Other notions of connectedness, Connectedness - Connectivity |  | | Connectedness, Connectedness - Connectivity, Connectedness - Other notions of connectedness, connected space, connected graph, connected category, connected component (graph theory), connected sum, connectivity (graph theory), simply connected, strongly connected component, totally disconnected |  | |
|  |  | 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.
Other related archivesLie groups, Mathematics, boundary, category, category theory, connected, connected category, connected component (graph theory), connected graph, connected space, connected sum, connectivity, connectivity (graph theory), contractible, directed graph, directed path, disk, graph theory, graphs, manifolds, mathematics, morphism, noun, open sets, ordered pair, path, path connected, set, simply connected, sphere, strongly connected, strongly connected component, topological space, topology, torus, totally disconnected, vertices
 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 |
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