A topological invariant important in algebraic topology and the first of the homotopy groups. For a path-connected space X with base point x₀, the fundamental group π‎₁(X) is the set of equivalence classes of continuous loops γ‎:[0,1]→X such that γ‎(0)=γ‎(1) = x₀, and where two loops are equivalent if they are homotopic. The product γ‎₁⁎γ‎₂ of two loops is the loop γ‎₂ followed by γ‎₁ and the inverse of γ‎ is the loop in reverse. A continuous function f:X→Y induces a homomorphism f_⁎:π‎₁(X)→π‎₁(Y) by sending γ‎ to f(γ‎). The circle’s fundamental group is ℤ; the knot group of a knot is the fundamental group of its complement.