If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither? If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither?
I think it is closed, is that right? How can I prove it?

If $A$ is an open subset of $B$, and both $B$ and $A$ are subsets of a space $X$, then $B\backslash A=(X\backslash A)\cap B$. As $A$ is open, $X\backslash A$ is closed, and so $B\backslash A$ is the intersection of two closed sets, and is closed.
