Definition:Metric Subspace

Definition
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 $\struct {A, d}$.

Also see

 * Subspace of Metric Space is Metric Space