A connected space is a topological space that cannot be written as the union of two nonempty, disjoint open sets (Sharifi, p. 39). Requiring any two points to be joined by a path instead defines pathwise-connectedness. Every pathwise-connected topological space is connected, but the converse fails. For example, the topologist's sine curve is connected but not pathwise-connected (Sharifi, p. 42).
A nonempty topological space is 0-connected (a.k.a. pathwise-connected) if every continuous
map from a 0-sphere to the topological
space extends continuously to the 1-disk. Since the
0-sphere consists of the two endpoints
of an interval (1-disk), every
two points have a path between
them. A topological space is 1-connected (a.k.a.
simply connected) if it is 0-connected and every
continuous map from the 1-sphere
to it extends continuously to a map from the 2-disk.
In other words, every loop in the topological
space is null-homotopic. For an integer 
, a topological
space is 
-connected
if it is 
-connected
and every continuous map from the 
-sphere into it extends continuously
over the 
-disk (Hatcher 2002, p. 346).
By Whitehead's theorem, a nonempty CW-complex is 
-connected
for every nonnegative integer 
iff it is contractible
(Hatcher 2002, p. 346).
