# connected

A space *S* is said to be connected if
any two points in *S* can be connected by a curve lying wholly within *S*. Two spaces can be added by what is called
a **connected sum**. Roughly speaking, this involves pulling
out a disk from each surface, creating holes,
and then sewing the two surfaces together along the boundaries of the holes.
In this way, a one-holed torus can be added
to a two-holed torus to give a three-holed torus; alternatively, a projective
plane can be added to a projective plane to give a Klein
bottle. The operation is commutative and associative and there is even an
identity element: for example, adding a sphere to any surface simply returns
the same surface.