Connected Space

A space  is connected if any two points in  can be connected by a curve lying wholly within .

A space is 0-connected (a.k.a. pathwise-connected) if every map from a 0-sphere to the space extends continuously to the 1-disk. Since the 0-sphere is the two endpoints of an interval (1-disk), every two points have a path between them. A space is 1-connected (a.k.a. simply connected) if it is 0-connected and if every map from the 1-sphere to it extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible. A space is -multiply connected if it is -connected and if every map from the -sphere into it extends continuously over the -disk.

A theorem of Whitehead says that a space is infinitely connected iff it is contractible.