Isolated Point in Metric Space iff Isolated Point in Topological Space

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $T = \struct {A, \tau}$ be the topological space with the topology induced by $d$.

Let $H \subseteq A$.

Let $x \in H$

Then:
 * $x$ is an isolated point of $H$ in $M$ $x$ is an isolated point of $H$ in $T$

Proof
From Open Balls form Local Basis for Point of Metric Space, the set:
 * $\BB_x = \set {\map {B_\epsilon} x : \epsilon \in \R_{>0} }$

is a local basis of $x$.

From Local Basis Test for Isolated Point:
 * $x$ is an isolated point of $H$ in $T$ $\exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \cap H = \set x$

By definition of an isolated point in $M$:
 * $x$ is an isolated point of $H$ in $T$ $x$ is an isolated point of $H$ in $M$