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.
