# When is a collection of sets closed under union?

I started studying Probability, and I'm not sure if I understand what is the meaning of "closed under union".

A collection (say $$F$$) of subsets of a set (say $$\Omega$$) is said to be a $$\sigma$$-algebra if:

• $$\Omega \in F$$
• $$F$$ is closed under complement
• $$F$$ is closed under union

Now, consider the following example:

Given $$\Omega = \{1, 2, 3, 4, 5\}$$, is $$F = \{\emptyset, \{1, 2\}, \{3, 4, 5\}, \{5\}, \{1, 2, 3, 4\}, \{1, 2, 3, 4, 5\}\}$$ closed under union?

I'm asking because I know that $$\bigcup_{i = 1}^{6} X_i \in F$$; however it does not happen for any two elements, for example: $$\{1,2\} \cup \{5\} \not\in F$$.

• $\sigma$-algebras are only required to be closed under countable unions. Jan 6 '19 at 8:42
• slight change to title; a collection of sets can be closed under union, not the set itself; Jan 6 '19 at 9:09
• You're right! Thanks, I changed it. Jan 6 '19 at 9:10

"$$F$$ is closed under union" means that for all $$A,B \in F$$, $$A \cup B \in F$$.
So here $$\{1,2\}$$ and $$\{5\}$$ are elements of $$F$$, but their union is not, so $$F$$ is not closed by union.
“Closed under union” means that the union of any set of members of $$F$$ is also a member of $$F$$.
In your example $$F$$ is not closed under union (although it is closed under complement).