The union of $A$ and $B$ is closed in $X$

Question

Say if $A$ and $B$ are non empty closed subsets of $(X,d)$.

am I right in them saying that $A\cup B$ is also closed in $X$?

Answer 1

$A^c, B^c$ are open ($c$ for complement).

$A^c \cap B^c$ is open (easy to prove).

Their complement

$(A^c \cap B^c)^c = (A\cup B)$

is closed.

Used: $(D^c)^c =D, D=A,B$, and

De Morgan's law.

Answer 2

For brevity, for a set $S$ I'll write $X \backslash S$ as $S^C$. $A$ and $B$ are closed in the metric space $X$. So $A^C$ and $B^C$ are open

De Morgan's law tells us that $(A \cup B)^C = (A^C \cap B^C) $. So $A \cup B$ is the complement of an open set, and is therefore closed.

This can easily be generalised to see the following: The union of finitely many closed sets is closed