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?

Yes, it is probably a typo. It should be $\emptyset=\mathbb{R}\setminus\mathbb R$.
