This is not a homework assignment, its a question from a topology test which im trying to solve to prepare for my own.
I dont know how to solve it, so anything helpful will be appreciated.
Let $(X,d)$ be a complete metric space and $\{G_i : i=1,2,\ldots\}$ a countable collection of open subsets such that $B=\bigcap G_i$ is not empty. Let $\{A_j : j=1,2,\ldots\}$ be a countable collection of nowhere dense sets in $X$.
a. Prove $B\subset \overline{B\setminus\bigcup A_j}$
b. Show by counter example that the condition of completeness on $(X,d)$ is necessary.
I know that a complete metric space is a Baire space, so any intersection of open dense sets is dense, also $X\setminus\overline{A_i}$ is dense in X, but i dont quite see how to apply it. Is $B\setminus\overline{A_i}$ dense in B?
The claim is false - take $ X = \mathbb R $, $ G_i = (-1/i, 1/i) $ and $ A_k = \{ 0 \} $ for all $ k $. Then,
$$ B = \bigcap G_i = \{ 0 \} $$ $$ C = B - \bigcup A_j = B - \{ 0 \} = \emptyset $$
so that $ \bar{C} = \emptyset $, and $ B $ is not a subset of $ \bar C $.