  For a path-connected metric space $(X,d)$, the $n$-th homotopy group
$\pi_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space
with the uniform metric. This pseudometric gives $\pi_n(X)$ the structure of a
topological group and when $X$ is compact, the induced pseudometric topology is
independent of the metric $d$. In this paper, we study the properties of this
pseudometric and how it relates to previously studied structures on $\pi_n(X)$.
Our main result is that the pseudometric topology agrees with the shape
topology on $\pi_n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse
limit of finite polyhedra with retraction bonding maps.
