TITLE: Understanding why the empty set is closed
QUESTION [12 upvotes]: 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?

REPLY [25 votes]: 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.)