is
path connected if for each pair of points
there
exists a continuous function
from
the unit interval
towith
and
University Maths Notes: Topology – Path Connectedness
A topological spaceis
path connected if for each pair of pointsthere
exists a continuous functionfrom
the unit intervaltowithand
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.