Finite Union of Closed Sets is Closed/Normed Vector Space

Theorem

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

Let $\ds \bigcup_{i \mathop = 1}^n F_i$ 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 $M$.

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

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

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

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

$\blacksquare$

2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces