Understanding why the empty set is closed 
Definition. A set is called closed if its complement in $\mathbb{R}$ is open.

In my lecture notes it says: $\emptyset$ is closed because $\emptyset = \emptyset \setminus \mathbb{R}$ and $\mathbb{R}$ is open. I think there is a typo because $\emptyset \neq \emptyset \setminus \mathbb{R}$, right? It should be $\emptyset = \mathbb{R} \setminus \mathbb{R}$. Can you please check this?
 A: The complement of $\emptyset$ in $\mathbb{R}$ is $\mathbb{R}\setminus \emptyset$ which is equal to $\mathbb{R}$. And $\emptyset$ is closed in $\mathbb{R}$ because $\mathbb{R}$ is open in $\mathbb{R}$. 
Furthermore, even though $\emptyset=\emptyset\setminus \mathbb{R}$, that doesn't let's us conclude that $\emptyset$ is closed in $\mathbb{R}$.
A: It's both correct and a typo.
That is:


*

*The useful statement is "$\mathbb{R}=\mathbb{R}\setminus\emptyset$": since $\mathbb{R}$ is open, this means the complement of $\emptyset$ (in $\mathbb{R}$) is open - so $\emptyset$ is closed. This is (presumably) what the author meant to write.

*However, it is true that $\emptyset=\emptyset\setminus\mathbb{R}$; it's just not helpful here. Remember that "$A\setminus B$" is the set of all things in $A$ which aren't in $B$. Well, there are no things in $\emptyset$ which aren't in $\mathbb{R}$ (in fact, there are no things in $\emptyset$ at all!), so $\emptyset\setminus\mathbb{R}=\emptyset$. (I'm pointing this out because you ask whether $\emptyset\setminus\mathbb{R}\not=\emptyset$, at the end of your question.)
A: Yes, it is probably a typo. It should be $\emptyset=\mathbb{R}\setminus\mathbb R$.
