What does it mean for a sigma algebra to be generated by something?

Question

I have a definition:

Given $C \supset 2^{\Omega}$, the $\sigma$- algebra generated by $C$ written, $\sigma(C)$ is the "smallest" $\sigma$- algebra containing $C$

I understand what this means but I just don't understand what "generated by $C$" means.

Similarly, I am given an example of the Borel $\sigma$- algebra as $\sigma(T)$ where $T = ${open sets of $\mathbb{R}$}

So a Borel $\sigma$- algebra is equal to a $\sigma$- algebra generated by all the open sets of $\mathbb{R}$

Can someone please explain what the word "generated by" means?

I know a $\sigma$- algebra is a collection of subsets of the power set $2^{\Omega}$ where $\Omega$ is any set. So does this mean that if a $\sigma$- algebra is generated by something else, that something else is just the set $\Omega$?

Sorry for such a basic question, just looking for some clarification.

Answer 1

You can give meaning to the "smallest" $\sigma$-algebra containing $C\subset2^\Omega$ by first noting that any intersection of $\sigma$-algebras on $\Omega$ is again a $\sigma$-algebra on $\Omega$ and then setting

$$ \sigma(C):=\bigcap_{\mathcal A \in \mathbb A}\mathcal A, \quad \text{where $\mathbb A:=\{\mathcal A\subset 2^\Omega : \text{$\mathcal A$ is a $\sigma$-algebra on $\Omega$ and $C\subset\mathcal A$}\}$.} $$

Answer 2

An intersection of $\sigma$-algebras is a $\sigma$-algebra, similar to the fact that an intersection of subgroups is a subgroup. This fact gives us that there is a unique minimal $\sigma$-algebra containing $C$ arising from a collection $C$, such that it generates the $\sigma$-algebra in that sense.

Which is again similar to what it means for a subgroup to be generated by a set, for a topology to be generated by a collection of sets and so on.

At least that's my take on it.