Intersection of elements of a sigma algebra with a common element.
Solution 1:
Define an equivalence relation $\sim$ on $\Omega$ such that $\omega\sim\omega'$ if and only if $\omega$ and $\omega'$ lie in exactly the same elements of $\mathcal{B}.$ Note that $A(\omega)$ is simply the equivalence class containing $\omega$. Let $\mathcal{F}$ consist of arbitrary unions of such equivalence classes. Verify that $\mathcal{F}$ is a $\sigma$-algebra containing $\mathcal{B}$ and that $$A(\omega) = \cap \{A:\, A\in \mathcal{F}, \, \omega \in A\}.$$