Subspace metric

Definition

Suppose $(X,d_X)$ is a 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.