Metric induces topology

Statement

Suppose $(X,d)$ is a metric space. Then, the collection of subsets:

$B(x,r):=\{y\in X\mid d(x,y)<r\}$

form a basis for a topology on $X$. These are often called the open balls of $X$.

Definitions used

Metric space

Further information: metric space

A metric space $(X,d)$ is a set $X$ with a function $d:X\times X\to \mathbb {R}$ satisfying the following:

• $d(x,y)\geq 0\ \forall \ x,y\in X$ (non-negativity)
• $d(x,y)=0\iff x=y$ (identity of indiscernibles)
• $d(x,y)=d(y,x)$ (symmetry)
• $d(x,y)+d(y,z)\geq d(x,z)\ \forall \ x,y,z\in X$ (triangle inequality)

Basis for a topological space

Further information: Basis for a topological space

A collection of subsets $\{U_{i}\}_{i\in I}$ of a set $X$ is said to form a basis for a topological space if the following two conditions are satisfied:

• $\bigcap _{i\in I}U_{i}=X$
• For any $i,j\in I$, and any $p\in U_{i}\cap U_{j}$, there exists $U_{k}\subset U_{i}\cap U_{j}$ such that $p\in U_{k}$.

Note that this is the definition for a collection of subsets that can form the basis for some topology.