The fundamental group of an arcwise-connected set X is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a given basepoint p, under the equivalence relation of homotopy. The identity element of this group is the set of all paths homotopic to the degenerate path consisting of the point p. The fundamental groups of homeomorphic spaces are isomorphic. In fact, the fundamental group only depends on the homotopy type of X. The fundamental group of a topological space was introduced by Poincaré.

Cayley graph | co-H-space | connected set | deck transformation | group | homology | homotopy group | Milnor's theorem | universal cover | van Kampen's theorem