What is the difference between definitions of open set and saturated set?
Solution 1:
No, it's not a synonym. For instance, in $\Bbb R$, endowed with the usual topology, the set $\{0\}$ is not an open set, but it is saturated, since$$\{0\}=\bigcap_{n\in\Bbb N}\left(-\frac1n,\frac1n\right).$$Note that, in a topological space, an arbitrary intersection of open sets doesn't have to be open.