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 Connectivity, Locally Pathwise-Connected Space, Multiply Connected, Pathwise-Connected, Simply Connected