Metric Space defined by Closed Sets

Theorem

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

Then:

\((\text C 1)\)	$:$		$A$ is closed in $M$
\((\text C 2)\)	$:$		$\O$ is closed in $M$
\((\text C 3)\)	$:$		The union of a finite number of closed sets of $M$ is a closed set of $M$
\((\text C 4)\)	$:$		The intersection of arbitrarily many closed sets of $M$ is a closed set of $M$

Proof

From Metric Space is Closed in Itself, $\text C 1$ holds.

$\Box$

From Empty Set is Closed in Metric Space, $\text C 2$ holds.

$\Box$

Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the union of a finite number of closed sets of $M$.

Then from De Morgan's laws:

$\ds A \setminus \bigcup_{i \mathop = 1}^n V_i = \bigcap_{i \mathop = 1}^n \paren {A \setminus V_i}$

By definition of closed set, each of the $A \setminus V_i$ is by definition open in $M$.

We have that $\ds \bigcap_{i \mathop = 1}^n \paren {A \setminus V_i}$ is the intersection of a finite number of open sets of $M$.

Therefore, by Finite Intersection of Open Sets of Metric Space is Open, $\ds \bigcap_{i \mathop = 1}^n \paren {A \setminus V_i} = A \setminus \bigcup_{i \mathop = 1}^n V_i$ is likewise open in $M$.

Then by definition of closed set, $\ds \bigcup_{i \mathop = 1}^n V_i$ is closed in $M$.

Thus $\text C 3$ holds.

$\Box$

Let $I$ be an indexing set (either finite or infinite).

Let $\ds \bigcap_{i \mathop \in I} V_i$ be the intersection of a indexed family of closed sets of $M$ indexed by $I$.

Then from De Morgan's laws:

$\ds A \setminus \bigcap_{i \mathop \in I} V_i = \bigcup_{i \mathop \in I} \paren {A \setminus V_i}$

By definition of closed set, each of $A \setminus V_i$ are by definition open in $M$.

We have that $\ds \bigcup_{i \mathop \in I} \paren {A \setminus V_i}$ is the union of a family of open sets of $M$ indexed by $I$.

Therefore, by definition of a topology, $\ds \bigcup_{i \mathop \in I} \paren {A \setminus V_i} = A \setminus \bigcap_{i \mathop \in I} V_i$ is likewise open in $M$.

Then by definition of closed set, $\ds \bigcap_{i \mathop \in I} V_i$ is closed in $M$.

Thus $\text C 4$ holds.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.11$