Unions and Intersections over a Countable Index Set in a σ-Algebra

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $\mathcal{F}$ be a $\sigma$-algebra on a set $X$, let $I$ be a countable set, and let $A:I\to\mathcal{F}$, $i\mapsto A_{i}$, be a map. Unions and intersections of the family $(A_{i})_{i\in I}$ are as in Indexed Families of Sets and Their Union, Intersection and Product (ii) and Indexed Families of Sets and Their Union, Intersection and Product (iii).

(i) $\bigcup_{i\in I}A_{i}\in\mathcal{F}$.

(ii) If $I\neq\emptyset$, then $\bigcap_{i\in I}A_{i}\in\mathcal{F}$.