A hyperplane in $\mathbb{R}$ is a singleton, according to my lecture notes. A hyperplane $S$ is defined as:
$S$ = $Z$ + $\{x\}$, where $x \in X$ and $Z$ is the $\supset$-maximal proper linear subspace of $X$.
$S$ = $Z$ + $\{x\}$ is a notation for: $S = \{ z + x: (z, x) \in Z$ x $\{x\}\}$
Now comes my doubts: a proper linear subspace of $\mathbb{R}$ is $\{0\}$.
If it's $\{0\}$, then the hyperplane S = $\{0\}$ + $\{x\}$ = $\{x + 0: (x,0) \in \{x\}$ x $\{0\} \}$ = $\{x: x \in \{x\}\}?$ Then it results that a hyperplane is a singleton? Is that right?
Third edit: deleted the part about empty sets.
If $Z=\emptyset$, then $Z\times\{x\}=\emptyset$ and consequently also $Z+\{x\}=\emptyset$, this is not what you are looking for.
The only nonempty subspaces of $\mathbb{R}$ are the zero space $\{0\}$ and $\mathbb{R}$ itself, of which only the first one is a proper subspace. According to your definition, $Z=\{0\}$ is thus a maximal element among the proper subspaces (in this case the unique maximal one, in general there is no uniqueness) and hence the only hyperplanes are $S=Z+\{x\}= \{0\}+\{x\}=\{x\}$, which are indeed singeltons.