Is the difference of an open and a closed set open?

Question

Suppose that the set $A$ is open and the set $B$ is closed. Then $A \setminus B$ is open.

I can show this for two cases:

• If $A \cap B=\emptyset$, then $A \setminus B=A$, which is obviously open.

• If $A \subseteq B$, $A \setminus B=\emptyset$, which is open by definition

How do I show this for the general case, ie where $A \cap B \neq \emptyset$ and $A \not\subseteq B$?

Answer 1

Since $A\setminus B=A\cap(X\setminus B)$, where $X$ is the whole space, and since $A$ and $X\setminus B$ are open, $A\setminus B$ is open.

Answer 2

Hint:

$A\setminus B=A\cap B^c$

Answer 3

Hint: If $B$ is closed, then its complement $B^c$ is open. $A \setminus B = A \cap B^c$.