Metric subspaces

By verifying the axioms, it can be shown that any subspace $Y \subset X$ of a metric space $(X,\rho)$ itself becomes a metric space by restricting the domain of $\rho$ to $Y \times Y$ . This conveniently provides metrics on any of the manifolds and varieties from Chapter 4 by simply using any $L_p$ metric on ${\mathbb{R}}^m$ , the space in which the manifold or variety is embedded.