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