\begin{definition}[Definition:Dedekind Cut/Definition 1]
Let $\struct {S, \preceq}$ be a totally ordered set.
A '''Dedekind cut''' of $\struct {S, \preceq}$ is a non-empty proper subset $L \subsetneq S$ such that:
:$(1): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$ ($L$ is a lower set in $S$)
:$(2): \quad \forall x \in L: \exists y \in L: x \prec y$
\end{definition}