A topological space is said to be *connected*
if it cannot be represented as the union of two disjoint, nonempty,
open sets. While this definition is rather elegant and general, if
is connected, it does not imply that a path exists between any
pair of points in thanks to crazy examples like the
*topologist's sine curve*:

Consider plotting . The part creates oscillations near the -axis in which the frequency tends to infinity. After union is taken with the -axis, this space is connected, but there is no path that reaches the -axis from the sine curve.

How can we avoid such problems? The standard way to fix this is to
use the path definition directly in the definition of connectedness.
A topological space is said to be *path
connected* if for all
, there exists a path such that
and
. It can be shown that if is path connected, then
it is also connected in the sense defined previously.

Another way to fix it is to make restrictions on the kinds of
topological spaces that will be considered. This approach will be
taken here by assuming that all topological spaces are manifolds. In
this case, no strange things like (4.8) can
happen,^{4.7} and the definitions of connected and path connected coincide
[451]. Therefore, we will just say a space is *connected*. However, it is
important to remember that this definition of connected is sometimes
inadequate, and one should really say that is *path
connected*.

Steven M LaValle 2012-04-20