I am looking at the following definition:
Let $\kappa$ be a regular uncountable cardinal and let $\mathcal F$ be a filter on a non-empty set $X$. We say that $\mathcal F$ is $\kappa$-complete if $\bigcap \mathcal A \in \mathcal F$ for every family $\mathcal A$ of size less than $\kappa$ of subsets of $X$.
Why does $\kappa$ have to be regular? How does the definition break if $\kappa$ is singular?
Suppose that $\cal F$ was a $\kappa$-complete filter, and $\kappa$ was singular with cofinality $\lambda$.
Then for every family of $<\kappa$ sets, the intersection was in $\cal F$, but there was a family $\{A_i\mid i<\kappa\}$ such that $\bigcap A_i\notin\cal F$.
Let $\langle i_\alpha\mid\alpha<\lambda\rangle$ be a cofinal sequence in $\kappa$. Now let $B_\alpha=\bigcap_{i<i_\alpha}A_i$. By our assumption that $\cal F$ is $\kappa$-complete, $B_\alpha\in\cal F$.
But note that $\bigcap B_\alpha=\bigcap A_i\notin\cal F$, which is a contradiction since $\lambda<\kappa$.