Difference between generator and the sigma algebra generated by this generator Suppose $X$ is any set and $\mathcal{F} \subseteq 2^X $. By definition, I have learnt that $\sigma( \mathcal{F} ) $ is the smallest $\sigma$-algebra that contains $\mathcal{F} $. I am trying to understand the difference between $\sigma( \mathcal{F} ) $ and $\mathcal{F} $.
With specific example, suppose $\mathcal{F} = \{ f(A) : A \in \mathcal{G} \} $ where $\mathcal{G} $ is a $\sigma$-algebra. I am trying to understand $\sigma( \mathcal{F} ) $ and $f$ is a real-valued function defined on $X$. Obviously, $\sigma( \mathcal{F} ) $ contains all the $f(A) $ for $A \in \mathcal{G} $. What else does it contain? Because we don't have $\sigma( \mathcal{F} ) = \mathcal{F} $.
 A: If $\mathcal F$ is a $\sigma$-algebra, then $\sigma(\mathcal F)=\mathcal F$. Otherwise $\mathcal F$ is a proper subset of $\sigma(\mathcal F)$.
For your example, $\mathcal F$ need not be a $\sigma$-algebra. Take for instance $X = \{a\}$ and $f(a)=0$. Then $$\mathcal F=\{f(\varnothing), f(\{a\})\}= \{\varnothing, \{0\}\},$$
whereas
$$\sigma(\mathcal F) = \{\varnothing, \{0\}, \mathbb R\setminus\{0\}, \mathbb R\}.$$
A: Consider, for example, in the real line $\Bbb R$: $${\frak U }= \{ \left]a,b\right[ \mid a,b \in \Bbb R, \,a<b\}.$$ Then $[0,1] \in \sigma(\frak U)$ but $[0,1] \not\in \frak U$. In general, the sigma-algebra itself is way bigger than its generator. 
The sigma-algebra contains countable unions of elements of the generator, as well as countable intersections, countable unions of countable intersections, etc.
In the specific example above (or in general, about the sigma-algebra generated by the topology of a topological space), we call any set who is a countable union of open sets a $G_\delta$ set. We call any set who is a countable intersection of closed sets a $F_\sigma$ set.
We go on: any countable intersection of $F_\sigma$s we call a $F_{\sigma \delta}$ set, any countable union of $G_\delta$s we call a $G_{\delta\sigma}$ set, etc.
The generator consists only of open sets, but the sigma-algebra contains all such classes.
