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.

$\blacksquare$


Sources

  • 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
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