Find a collection $\frak U$ on $\mathcal P(X)$ closed by arbitrary unions which is not a topology on $X$.

Question

Given a set $X$ I am searching for a collection $\frak U$ on $\mathcal P(X)$ which satisfies the following two statements:

1. if $\cal U$ is contained in $\frak U$ then $\bigcup\cal U$ is in $\frak U$;
2. there exists $\cal U_\cap$ in $\frak U$ with $U_1$ and $U_2$ such that $U_1\cap U_2$ is not in $\frak U$.

So, I think that this counterexample (provided it exists) is relevant because otherwise joining $X$ to $\frak U$ then we can always generate a topology and so this would clarify definitively why topology is defined to be a collection closed even by finite intersections and not only by arbitrary unions. Could someone help me, please?

Answer 1

Assume $X$ has at least three points, $x, y, z$. Then the collection of all subsets that contain at least one of $x$ and $y$ is closed under arbitrary unions, but $\{ x, z \} \cap \{ y, z \}= \{ z \}$ is not in that collection so it's not closed under finite intersections.