Definition:Metric Subspace

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

Let $$Y \subseteq X$$.

Let $$d_Y: Y \times Y \to \reals$$ 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 $$\left({X, d}\right)$$.