Isolated Point in Metric Space iff Isolated Point in Topological Space
Jump to navigation
Jump to search
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 $M$
$\blacksquare$