Equivalence of Definitions of Topology Induced by Metric

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Theorem

The following definitions of the concept of Topology Induced by Metric are equivalent:

Definition 1

The topology on the metric space $M = \left({A, d}\right)$ 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 $M = \left({A, d}\right)$ 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 $M = \left({A, d}\right)$ be a metric space whose metric is $d$.


$(1)$ implies $(2)$

Let $T = \left({A, \tau_d}\right)$ be the topological space of which $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.

Then by definition:

$\tau_d$ is the set of all open sets of $M$.

...


Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 2.

$\Box$


$(2)$ implies $(1)$

Let $T = \left({A, \tau_d}\right)$ 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 $M$.

...

Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.