# 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:

- 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