Equivalence of Definitions of Closed Set in Metric Space

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Theorem

The following definitions of the concept of Closed Set in the context of Metric Spaces are equivalent:

Let $M = \left({A, d}\right)$ be a metric space.

Let $H \subseteq A$.

Definition 1

$H$ is closed (in $M$) if and only if its complement $A \setminus H$ is open in $M$.

Definition 2

$H$ is closed (in $M$) if and only if every limit point of $H$ is also a point of $H$.


Proof

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

Definition 1 implies Definition 2

Let $H$ be closed in $M$ by definition 1.

Then by definition $A \setminus H$ is open in $M$.

Let $b \in M: b \in A \setminus H$.

Then by definition of open set:

$\exists \delta \in \R_{>0}: B_\delta \left({b; d}\right) \subseteq A \setminus H$

where $B_\delta \left({b; d}\right)$ denotes the open $\delta$-ball of $b$.

From Intersection with Complement is Empty iff Subset:

$B_\delta \left({b; d}\right) \cap H = \varnothing$

So by definition of limit point:

$b \notin H'$

Hence:

$A \setminus H \cap H' = \varnothing$

By Intersection with Complement is Empty iff Subset:

$H' \subseteq H$

Thus $H$ contains all its limit points.

That is, $H$ be closed in $M$ by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $H$ be closed in $M$ by definition 2.

That is:

$H' \subseteq H$

Then by Set Complement inverts Subsets:

$A \setminus H \subseteq A \setminus H'$

Let $b \in A \setminus H$.

Then by definition of relative complement:

$b \notin H'$

Then by definition of limit point:

$\exists \delta \in \R_{>0}: B_\delta \left({b; d}\right) \cap H = \varnothing$

and so:

$B_\delta \left({b; d}\right) \subseteq A \setminus H$

As $b$ is arbitrary, it follows that $A \setminus H$ is open in $M$.

Hence by definition, $H$ is closed in $M$ by definition 1.

$\blacksquare$


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