Densely Ordered/Examples/Arbitrary Non-Densely Ordered

Example of Ordered Set which is not Densely Ordered
Let $S$ be the subset of the rational numbers $\Q$ defined as:
 * $S = \Q \cap \paren {\hointl 0 1 \cup \hointr 2 3}$

Then $\struct {S, \le}$ is not a densely ordered set.

Thus $\struct {S, \le}$ is not isomorphic to $\struct {\Q, \le}$.

Proof
It will be noted that $1 \in S$ and $2 \in S$ but there exists no $c \in S$ such that $1 < c < 2$.

Hence the result from Densely Ordered is Order Property.