\begin{definition}[Definition:Metric Subspace]
Let $\struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$.
That is, let $\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$.
Then $d_H$ is the '''metric induced on $H$ by $d$''' or the '''subspace metric of $d$ (with respect to $H$)'''.
The metric space $\struct {H, d_H}$ is called a '''metric subspace of $\struct {A, d}$'''.
\end{definition}