Let $A,B$ be subsets of a topological space $X$. Is it true that $\overline{A}-\overline{B}\subseteq\overline{A-B}$?
Suppose $x\in\overline{A}-\overline{B}$. So all open sets containing $X$ also contains an element of $A$. And there exists an open set $U$ containing $X$ that contains no element of $B$. So $U$ contains an element of $A-B$. But this is not enough to conclude that $x\in\overline{A-B}$. So I'm thinking the answer might be negative, but cannot find a counterexample.
Best Answer
If $x\notin\operatorname{cl}(A\setminus B)$, then $x$ has an open nbhd $U$ such that $U\cap(A\setminus B)=\varnothing$. And $$U\cap(A\setminus B)=(U\cap A)\setminus(U\cap B)\;,$$ so $U\cap A\subseteq U\cap B$. If $x\notin\operatorname{cl}B$, then $x$ has an open nbhd $V$ such that $V\cap B=\varnothing$. Let $W=U\cap V$. Then
$$W\cap A=W\cap(U\cap A)\subseteq W\cap(U\cap B)=W\cap B=\varnothing\;,$$
so $W\cap A=\varnothing$, and $x\notin\operatorname{cl}A$. Thus, if $x\in(\operatorname{cl}A)\setminus\operatorname{cl}B$, then $x\in\operatorname{cl}(A\setminus B)$.