# Isolated Point in Metric Space iff Isolated Point in Topological Space

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## 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 $M$ if and only if $x$ 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 $T$ if and only if $\exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \cap H = \set x$

By definition of an isolated point in $M$:

- $x$ is an isolated point of $H$ in $T$ if and only if $x$ is an isolated point of $H$ in $M$

$\blacksquare$