Equivalence of Definitions of Topology Induced by Metric
Theorem
The following definitions of the concept of Topology Induced by Metric are equivalent:
Definition 1
The topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$ is defined as the set $\tau$ of all open sets of $M$.
Definition 2
The topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$ is defined as the topology $\tau$ generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.
Proof
Let $M = \struct {A, d}$ be a metric space whose metric is $d$.
$(1)$ implies $(2)$
Let $T = \struct {A, \tau_d}$ be the topological space of which $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.
Then by definition:
Let $U \in \tau_d$.
From definition:
- $\forall y \in U: \exists \epsilon_y \in \R_{>0}: \map {B_{\epsilon_y} } y \subseteq U$
where $\map {B_{\epsilon_y}} y$ is the open $\epsilon$-ball of $y$.
We find these $\epsilon$ for every $y \in U$.
Consider the set $\ds \bigcup_{y \mathop \in U} \map {B_{\epsilon_y} } y$.
From $y \in \map {B_{\epsilon_y} } y \subseteq U$ for each $y$:
- $\ds \bigcup_{y \mathop \in U} \map {B_{\epsilon_y} } y = U$
This shows that any $U \in \tau_d$ is a union of open $\epsilon$-balls in $M$.
So open $\epsilon$-balls in $M$ form a basis for $\tau_d$.
Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 2.
$\Box$
$(2)$ implies $(1)$
Let $T = \struct {A, \tau_d}$ be the topological space of which $\tau_d$ is the topology induced on $M$ by $d$ by definition 2.
Then by definition:
- $\tau$ is the topology generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.
Let $U \in \tau_d$.
Since open $\epsilon$-balls in $M$ form a basis for $\tau_d$, $U$ is a union of open $\epsilon$-balls in $M$.
Hence for each $x \in U$:
- $\exists y \in U: \exists \epsilon \in \R_0: x \in \map {B_\epsilon} y$
By Open Ball is Subset of Open Ball:
- $x \in \map {B_{\epsilon - \map d {x, y} } } x \subseteq \map {B_\epsilon} y \subseteq U$
Therefore $U$ is an open set of $M$.
Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.
$\blacksquare$