Densely Ordered/Examples

Examples of Densely Ordered Sets

Arbitrary Non-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 $\struct {\Q, \le}$.

Arbitrary Densely Ordered

Let $S$ be the subset of the rational numbers $\Q$ defined as:

$S = \Q \cap \paren {\openint 0 1 \cup \hointr 2 3}$

Then $\struct {S, \le}$ is a densely ordered set.

Hence $\struct {S, \le}$ is isomorphic to $\struct {\Q, \le}$.