I am currently having a fundamental misunderstanding regarding $\sigma$-algebras. In particular, I am a bit confused about a specific example of the "smallest" $\sigma$-algebra on $\mathbf{R}$ that contains the set of subsets $\mathcal{A}=\{(0,1), (0,\infty)\}$. According to my text, the smallest $\sigma$-algebra on $\mathbf{R}$ that contains $\mathcal{A}$ is given by $$\mathcal{S}=\{\emptyset, (0,1), (0,\infty), (-\infty,0]\cup[1,\infty), (-\infty,0], [1,\infty), (-\infty,1), \mathbf{R}\}.$$
My question here is somewhat of a silly notational one regarding all the sets at play here. When I think of $\mathcal{S}$ containing $\mathcal{A}$, I think that the entire set $\{(0,1),(0,\infty)\}$ should be explicitly found within $\mathcal{S}$, i.e. $\mathcal{S}=\{\{(0,1),(0,\infty)\}, ...\}$. Instead, all I see in $\mathcal{S}$ are the individual elements of $\mathcal{A}$, and not $\mathcal{A}$ itself, if that makes sense.
Honestly I have no idea how people study this stuff, because it's the little things like this that just stop me dead in my tracks! Can anybody explain where I am going wrong here?
Thanks as always!