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?

Given $A$ is open and $B$ is closed.
\begin{eqnarray*}
B-A&=&\{x|x\in B \ \& \ x\notin A\}\\
&=&\{x|x\in B \ \& \ x\in A^c\}\\
&=&\{x|x\in B\cap A^c\}\\
B-A&=&B\cap A^c
\end{eqnarray*}
which is closed since it is the intersection of closed sets.
