Is an empty set clopen or neither

Question

I thought the set $\mathbb{R}$ is clopen, so its complement, $\emptyset$, is neither open nor closed. But from my research, it seems $\emptyset$ is also clopen. Can someone explain why?

And if a set has only one element, it's closed, right?

Answer 1

It is part of the axioms of a topology that the empty set is open and that the whole space is open. Since a closed set is per definition a set whose complement is open and since the complement of the empty set and the whole space is the respective other, this immediatly yields that both the emtpy set and the whole space are clopen.

Regarding your second question, it is not necessarily true that every one element set is closed. Consider for example $\mathbb{R}$ with the topology $\mathcal{O}=\{\emptyset, \mathbb{R} \}$. Nevertheless the condition is quite important and is called $T_1$.