Definition:Metric Subspace

Definition

Let $\left({X, d}\right)$ be a metric space.

Let $Y \subseteq X$.

Let $d_Y: Y \times Y \to \R$ be the restriction $d \restriction_{Y \times Y}$ of $d$ to $Y$.

That is, let $\forall x, y \in Y: d_Y \left({x, y}\right) = d \left({x, y}\right)$.

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

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

The metric space $\left({Y, d_Y}\right)$ is called a metric subspace of $\left({X, d}\right)$.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Example $2.2.5$