Definition:Densely Ordered

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Then $\struct {S, \preceq}$ is defined as densely ordered if and only if strictly between every two elements of $S$ there exists another element of $S$:

$\forall a, b \in S: a \prec b \implies \exists c \in S: a \prec c \prec b$

where $a \prec b$ denotes that $a \preceq b$ but $a \ne b$.

Densely Ordered Subset

A subset $T \subseteq S$ is said to be densely ordered in $\struct {S, \preceq}$ if and only if:

$\forall a, b \in S: a \prec b \implies \exists c \in T: a \prec c \prec b$

Also known as

The term close packed is also used for densely ordered.

Some sources merely use the term dense.

Examples

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 $\struct {\Q, \le}$.

Also see

Compare with the topological concepts:

• Definition:Dense-in-itself
• Definition:Everywhere Dense
• Definition:Nowhere Dense

• Results about densely ordered sets can be found here.

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