4. Topology of Metric Spaces

Topology of Metric Spaces

Metric Spaces and Their Topology

Metric Space: A metric space is a set $X$ equipped with a distance function $d:X\times X\to \mathbb{R}$ satisfying the following properties for all $x,y,z$ in $X$:

1. Non-negativity: $d(x,y)\ge 0$ with equality if and only if $x=y$.
2. Identity of indiscernibles: $d(x,y)=0$ if and only if $x=y$.
3. Symmetry: $d(x,y)=d(y,x)$.
4. Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$.

The topology induced by the metric $d$ consists of open sets defined by open balls $B(x,r)=\{y\in X\mid d(x,y)<r\}$ for every $x$ in $X$ and $r>0$.

Properties of Metric Spaces

Completeness: A metric space $X$ is complete if every Cauchy sequence in $X$ converges to a limit in $X$.

Compactness: A metric space $X$ is compact if every open cover of $X$ has a finite subcover.

Connectedness: A metric space $X$ is connected if it cannot be divided into two disjoint nonempty open sets.

Examples of Metric Spaces

Euclidean Space: The Euclidean space ${\mathbb{R}}^n$ equipped with the standard Euclidean metric is a metric space. The distance between two points $x=(x_1,x_2,\dots ,x_n)$ and $y=(y_1,y_2,\dots ,y_n)$ is given by $d(x,y)=\sqrt{(x_1-y_1{)}^2+(x_2-y_2{)}^2+\dots +(x_n-y_n{)}^2}$.

Discrete Metric Space: Any set $X$ equipped with the discrete metric $d(x,y)=\{\begin{array}{ll}0& \text{if }x=y\\ 1& \text{if }x\ne y\end{array}$ is a metric space.