Fundamental Group

The fundamental group of a Connected Set $S$ is the Quotient Group of the Group of all paths with initial and final points at a given point $P$ and the Subgroup of all paths Homotopic to the degenerate path consisting of the point $P$.

The fundamental group of the Circle is the Infinite Cyclic Group. Two fundamental groups having different points $P$ are Isomorphic. If the fundamental group consists only of the identity element, then the set $S$ is Simply Connected.

See also Milnor's Theorem