Subspace metric

Definition
Suppose $$(X,d_X)$$ is a defining ingredient::metric space, and $$Y \subseteq X$$. The subspace metric on $$Y$$ is defined by simply restricting the metric on $$X$$ to points in $$Y$$. In other words, for $$a,b \in Y$$, we define $$d_Y(a,b) = d_X(a,b)$$.

Note that if we start with a geodesic metric space, the subspace metric on a subspace need not be a geodesic metric any more.