Definition:Densely Ordered/Subset

Definition

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

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

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

Warning

An intuitive understanding of the concept of densely ordered subset is likely to lead you astray.

Note that, for example, the closed real interval $\closedint 0 1$ is not densely ordered in $\R$.

This is because, while $\R$ is densely ordered in itself, and so is $\closedint 0 1$, the elements $2$ and $3$ in $\R$ have no elements of $\closedint 0 1$ between them.

This is just how the definition is constructed.

Also known as

The term close packed is also used for densely ordered.

Some sources merely use the term dense.

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

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