Isolated Point in Metric Space iff Isolated Point in Topological Space

From ProofWiki
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 $M$

$\blacksquare$