Is an empty set clopen or neither? 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?
 A: 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 $T_1$.
A: Since a set $A$ is closed if and only if $\Bbb R\setminus A$ is open, it follows from the fact that $\Bbb R$ is clopen that $\emptyset$  is clopen too.
A: $\emptyset$ and $\Bbb R$ are open by definition of a topology, and as $\emptyset$ is the complement of $\Bbb R$ it is also closed, and $\Bbb R$ is the complement of $\emptyset$ so it is also closed. So both sets are clopen in any topology on $\Bbb R$.
In a metric topology, like the usual topology, indeed all sets of the form $\{x\}$ are closed.
If a topology on $\Bbb R$ has the property that the only clopen (closed-and-open) subsets are $\emptyset$ and $\Bbb R$, we call this topology "connected". The usual topology on $\Bbb R$ has this property.
A: In $\mathbb{R^n}$ a set is open if and only if it contains none of its boundary points.
Further, in $\mathbb{R^n}$ a set is closed if and only if it contains all of its boundary points.
Therefore, in $\mathbb{R^n}$, the only sets that can be simultaneously open and closed, are sets that have no boundary points.
The only such sets are $\mathbb{R^n}$ itself, and $\emptyset$.
