Suppose $A$ is an open set and $B$ a closed set. Is it necessarily true that $A \backslash B$ open? I have thought about a few counter examples but could not come up with any, so I am inclined to say it is true. For example, $\mathbb{R} \backslash \phi$ is both open and closed but this does not impose any problem.
If $A \subset B$ then $A \backslash B = \phi$ which is open. If $A$ and $B$ do not intersect each other then $A \backslash B = A$ which is of course open. The only edge case is when $B \subset A$, which again I think should make $A \backslash B$ open. Am I right?
Best Answer
ANSEWR ACCORDING COMMENT:
Take $A=(-2,2)$ and $C_n=[-1+1/n,1-1/n]$.
Then $B=\bigcup C_n=(-1,1)$ and $A\setminus B=(-2,-1]\cup[1,2)$.
FIRST ANSWER:
You're not right. You missed the genereal case when $A\cap B\ne\varnothing$ but neither $A\subset B$ nor $B\subset A$.
A better approach: Recall that $A\setminus B=A\cap B^c$ and take into account that intersection of finite number of open set is always open.