Definition:Metric Subspace

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

Let $$Y \subseteq X$$.

Let $$d^{\prime}$$ be the restriction of $$d$$ to $$Y$$.

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

Then $$d^{\prime}$$ is a metric on $$Y$$ and is referred to as the metric induced on $$Y$$ by $$\left\{{X, d}\right\}$$.

The metric space $$\left\{{Y, d^{\prime}}\right\}$$ is called a metric subspace of $$\left\{{X, d}\right\}$$.