# Finite Union of Closed Sets is Closed

## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

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

## Proof

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

Then from De Morgan's laws:

- $\displaystyle S \setminus \bigcup_{i \mathop = 1}^n V_i = \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right)$

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

We have that $\displaystyle \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right)$ is the intersection of a finite number of open sets of $T$.

Therefore, by definition of a topology, $\displaystyle \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right) = S \setminus \bigcup_{i \mathop = 1}^n V_i$ is likewise open in $T$.

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

$\blacksquare$

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous) ... (next): $2.24 d$ - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.2$: Topological Spaces: Exercise $4$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$: Theorem $1$