Set of Isolated Points of Metric Space is Disjoint from Limit Points

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Theorem

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

Let $H \subseteq A$ be a subset of $A$.

Let $H'$ be the set of limit points of $H$.

Let $H^i$ be the set of isolated points of $H$.


Then:

$H' \cap H^i = \O$


Proof

Let $a \in H_i$.

Then by definition of isolated point:

$\exists \epsilon \in \R_{>0}: \set {x \in H: \map d {x, a} < \epsilon} = \set a$

But by Metric Space Axiom $\text M 1$:

$\map d {a, a} = 0$

and so:

$\set {x \in H: 0 < \map d {x, a} < \epsilon} = \O$

So by definition $a$ is not a limit point of $H$.

That is:

$a \notin H'$

or:

$a \in \relcomp A {H'}$

It follows from Intersection with Complement is Empty iff Subset that:

$H' \cap H^i = \O$

$\blacksquare$


Sources