# Definition(s) > [!NOTE] Definition (Metric) > A *metric* $d$ on a set $X$ is a [[Function|function]] $d:X\times X\to[0,\infty)$ which is > > 1. Non-degenerate: $\forall x,y\in X$ $d(x,y)=0 \iff x=y$ > 2. Symmetric: $\forall x,y\in X,$ $d(x,y)=d(y,x)$ > 3. Satisfies the triangle inequality: $\forall x,y,z\in X,$ $d(x,y)\leq d(x,z)+d(z,y)$ > [!Example] Example > By [[Euclidean Metric on Real n-Space is a Metric]], $d:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ defined by $d(x,y)=\lvert \lvert x-y \rvert \rvert$ is a metric on $\mathbb{R}^n .$ # Properties(s) # Application(s) **More examples**: # Bibliography