The set $\{ x \mid d(x,c) = r \}$ is called the sphere of radius $r$ with centre $c$ . This generalizes the notion of spheres to metric spaces.

Note that the sphere in a metric space need not look like a sphere in Euclidean space. For instance, if we impose the metric $d(x,y) = max \{|x_1-y_1|, |x_2-y_2|, |x_3-y_3|\}$ on $\mathbb{R}^3$ instead of the Euclidean metric, spheres according to this metric are actually cubes! Even more bizarre situations can occur in general -- a sphere might be disconnected, or it may be discrete, or it may even be an empty set.