Finite Union of Closed Sets is Closed/Normed Vector Space

Theorem
Let $M = \struct {X, \norm {\, \cdot \, }}$ be a normed vector space.

Then the union of finitely many closed sets of $M$ is itself closed.

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

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

Then from De Morgan's laws:


 * $\displaystyle X \setminus \bigcup_{i \mathop = 1}^n F_i = \bigcap_{i \mathop = 1}^n \paren {X \setminus F_i}$

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

By Finite Union of Open Sets in Normed Vector Space is Open, $\displaystyle \bigcap_{i \mathop = 1}^n \paren {X \setminus F_i} = X \setminus \bigcup_{i \mathop = 1}^n F_i$ is likewise open in $M$.

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