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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.
See also Connectivity, Locally Pathwise-Connected Space, Multiply Connected, Pathwise-Connected, Simply Connected