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?
Yes, if $A$ is open and $B$ is closed, then $B\setminus A$ is closed. To prove it, just note that $X\setminus A$ is closed (where $X$ is the whole space), and $B\setminus A=B\cap(X\setminus A)$, so $B\setminus A$ is the intersection of two closed sets and is therefore closed.

Alternatively, you can observe that $X\setminus(B\setminus A)=(X\setminus B)\cup A$ is the union of two open sets and therefore open, so its complement, $B\setminus A$, is closed.
Given $A$ is open and $B$ is closed.
\begin{eqnarray*}
B-A&amp;=&amp;\{x|x\in B \ \&amp; \ x\notin A\}\\
&amp;=&amp;\{x|x\in B \ \&amp; \ x\in A^c\}\\
&amp;=&amp;\{x|x\in B\cap A^c\}\\
B-A&amp;=&amp;B\cap A^c
\end{eqnarray*}
which is closed since it is the intersection of closed sets.