Definition:Metric Subspace

Definition
Let $\left({A, d}\right)$ 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 $\forall x, y \in H: d_H \left({x, y}\right) = d \left({x, y}\right)$.

The metric space axioms hold as well for $d_H$ as they do for $d$.

Then $d_H$ is a metric on $H$ and is referred to as the metric induced on $H$ by $d$ or the subspace metric of $d$ (with respect to $H$).

The metric space $\left({H, d_H}\right)$ is called a metric subspace of $\left({A, d}\right)$.