Metric

Definition

$d:X\times X\to {\mathbb{R}}_0^{+}$ Metric defined on $X$ is a function satisfying conditions:

• Positivity: $\forall x,y\in X,d(x,y)\ge 0\text{and}(d(x,y)=0\iff x=y)$
• Symmetry: $\forall x,y\in X,d(x,y)=d(y,x)$
• Triangle inequality: $\forall x,y,z\in X,d(x,y)+d(y,z)\ge d(x,z)$

Given a metric $d$ on $X$, $d(x,y)$ is called the distance between $x$ and $y$.

Distance Between Point and Set

Consider a Metric Space $(X,\mathcal{T},d)$, a point $x$ and a non-empty subset $A\subset X$. The distance from $x$ to $A$ is defined by Infimum $d(x,A):=\inf \{d(x,a)|a\in A\}$

Distance Between Sets

Consider a Metric Space $(X,\mathcal{T},d)$, and non-empty subsets $A,B\subset X$. The distance from $A$ to $B$ is defined by Infimum $d(A,B):=\inf \{d(a,b)|a\in A,b\in B\}$

Examples

Examples on Real Plane

Consider a set $X={\mathbb{R}}^2$, and points $x=(x_1,x_2)$ and y = (y_{1}, y_{2})

• Discrete metric: $d_1(x,y)=\{\begin{array}{ll}1& x\ne y\\ 0& x=y\end{array}$
• Euclidean metric: $d_2(x,y)=\sqrt{(x_1-y_1{)}^2+(x_2-y_2{)}^2}$
• Square metric: $d_3(x,y)=\max (|x_1-y_1|,|x_2-y_2|)$
• Standard bounded metric: $d_4(x,y)=\min (d_2(x,y),1)$

The metric topologies generated by the metrics have the following relation: $2^{\mathbb{R}}={\mathcal{T}}_{d_1}⊋{\mathcal{T}}_{d_2}={\mathcal{T}}_{d_3}={\mathcal{T}}_{d_4}$

Facts

a metric can be induced by a Norm $d(v,w)=\Vert v-w\Vert$