Set $A$ over $R := \{(−\infty, a], (a, b], (b, \infty), \emptyset, R, (a, b]\cup (b, \infty), (−\infty, a]\cup(a, b],(−\infty, a]\cup(b, \infty)\}$.
Set $B$ over $[0,10] := \{ [0,6], [4,10], \emptyset, [0,10], (4,6), [6,10], [0,4], [6,10] \cup [0,4] \}$.
They are both algebras, I think they are also $\sigma$ - algebras, but I am not 100% sure. They are closed under finite intersections. However, a countable set is either a finite set or a countably infinite set. so countable intersections implies finite intersections.