Figure: Simply connected and non- simply connected sets in
(ii)
In the interior of a sphere is simply connected. Interior of two concentric spheres is also simply
connected. Torus is not simply connected. For example, in the figure below, the curve is not the boundary of any surface in .
Figure: Simply connected and non- simply connected sets in
47.2.11
Note:
A more mathematically rigorous definition of simple connectedness is the following. Let be connected. We say is simply connected if given any two points and in and any two curves and in ,
the interior of a sphere is simply connected. Interior of two concentric spheres is also simply
is not the boundary of any surface in 
. 
be connected. We say 
is simply connected if given any two points 
and 
in 
and any two curves 
and 
in 
, 
