I am somehow confused about this exercise.
First of all, a definition: A system of subsets $\mathcal{R}\subset\mathcal{P}(\Omega)$, where $\Omega$ is a nonempty set $\Omega$, is called $\alpha$-system, if it does satisfy the following conditions:
- $\Omega\in\mathcal{R}$
- $A\in\mathcal{R}\Rightarrow A^c\in\mathcal{R}$
- For each sequence of subsets $A_1\subset A_2,...$ of elements from $\mathcal{R}$ it is true that $\bigcup_{n=1}^\infty A_n\in\mathcal{R}$
My goal is to prove, that for each subset $\mathcal{E}\subset\mathcal{P}(\Omega)$ it is true that there exists $\alpha(\mathcal{E})$, which the smallest $\alpha$-system which contains $\mathcal{E}$.
My thoughts: If I take a subset $\mathcal{E}\subset\mathcal{P}(\Omega)$, then I just have to
- add $\Omega$ and the empty set $\emptyset$ (if they are not already in $\mathcal{E}$)
- add each complement
- and make sure, that each union of a sequence of subsets $A_1\subset A_2,...$ of elements from $\alpha(\mathcal{E})$ is in $\alpha(\mathcal{E})$.
Concerning point three I have no ideas how to approach it. Any inspiration?