Multiply Connected

A set which is Connected but not Simply Connected is called multiply connected. 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, Pathwise-Connected, Simply Connected