Finite Union of Closed Sets is Closed

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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 $T$.

$\blacksquare$


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