Understanding why the empty set is closed

Solution 1:

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.)

Solution 2:

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 $\mathbb{R}$.