Metric induces topology

Statement

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

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

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 $d : X \times X \to R$ satisfying the following:

$d (x, y) \geq 0 \forall x, y \in X$ (non-negativity)
$d (x, y) = 0 ⟺ x = y$ (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:

$⋂_{i \in I} U_{i} = X$
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.