is the set F = {A|A or A^c is finite} over R a sigma algebra?

define $\displaystyle F = \{ A | A \mbox{ or } A^{c} \mbox{is finite} \} (\Omega = \mathbb{R}). \mbox{ Is F a } \sigma \mbox{-algebra over } \mathbb{R} ?$

My anwser: Yes as

a) if empty set = $\displaystyle A^{c}$ then A = $\displaystyle \mathbb{R} = \Omega$ which belongs to F

b) WLOG if A is finite let B = $\displaystyle A^{c}$ is uncoutnable but $\displaystyle B^{c}$ is fintire thus B belongs in F.

c) a union of finite sets is finite thus belong in F