Path Connectedness
A topological space
is path connected if for each pair of points
there exists a continuous function
from the unit interval
towith
and
Path connectedness is an equivalence relation:
since the constant map
is continuous.
since if
is continuous with
we can write
then
and
with
continuous by continuity of
and
since if there exists
and
with
then we can define
is continuous since
and
so there exists a path infrom
to
and
This equivalence relation splitsinto path connected components.
Every path-connected space is connected. The converse is not always true. The complex plane with infinity included is not path connected since
but there is no continuous function in
with
but
is path connected however.
Subsets of the real line
are connected if and only if they are path-connected; these subsets are the intervals of
Also, open subsets of
or
are connected if and only if they are path-connected. Connectedness and path-connectedness are the same for finite topological spaces.
An open set may or may not be path connected, but the closure of an open connected set is also path connected, since all closure does is include the limit points. Path connectedness is a topological property preserved by homeomorphisms. In particular, a homeomorphisms preserves the number of path connected components.
