Connected Space

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

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 n-multiply connected if it is (n-1)-connected and if every map from the n-sphere into it extends continuously over the (n+1)-disk.

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

See also

Connected Set, Contractible, Locally Pathwise-Connected, Multiply Connected, Pathwise-Connected, Simply Connected

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Cite this as:

Weisstein, Eric W. "Connected Space." From MathWorld--A Wolfram Web Resource.

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