Let $(X,d)$ be a metric space and $A_j \subseteq X$ for $j = 1,2,...$ . Let $B = \bigcup_{j=1}^\infty A_j$.
Find an example for which $\bigcup_{j=1}^\infty \bar{A_j} \neq \bar{B}$.
Also, in general what $A_j$ needs to be so that $\bigcup_{j=1}^\infty \bar{A_j}$ is a proper set of $ \bar{B}$.