Simply connected

is simply connected if

• it is path connected
• its fundamental group is trivial in some (hence all) basepoints.

Lemma

simply connected For each pair of points there is a unique homotopy class of paths between them.

Proof

Suppose is simply connected. Let . is path connected so have path . Suppose . Then is a loop based at .
. So relative to the endpoints
as loops based at
Existence of paths between any 2 points implies is path connected. Suppose is a loop based at . The loop is homotopic to the constant loop at so .