Pfadraum

Pfadraum, in topology often called the path space, is the space of all paths in a given topological space X. More precisely, for I = [0,1], the free path space is PX = C(I,X), the set of all continuous functions γ: I → X, equipped with the compact-open topology. A related construction is the based path space Px0X = { γ ∈ PX | γ(0) = x0 }, where x0 ∈ X is a chosen base point.

The based path space has a standard contractibility property. There is a natural homotopy H: Px0X × [0,1] → Px0X given by H(γ,t)(s) = γ(ts). This deforms any path to the constant path at γ(0) = x0, showing that Px0X is contractible.

There is a fundamental role for Pfadräume in fibrations. The evaluation map ev1: Px0X → X, defined by ev1(γ) = γ(1), is a (Serre) fibration. Its fiber over a point x ∈ X is the space of paths from x0 to x. This yields a canonical fibration sequence ΩX → Px0X → X, where ΩX is the based loop space (paths with endpoints at x0). Consequently, the homotopy groups satisfy πn(X) ≅ πn−1(ΩX) for n ≥ 2, and π1(X) ≅ π0(ΩX).

In practice, Pfadräume provide a convenient model for studying homotopy and fundamental groups. They also offer a functional-analytic viewpoint on X, since PX can be viewed as a function space XI with the compact-open topology, and, in many contexts, PX and related spaces contribute to constructions such as universal covers and classifying spaces.