Definition:Densely Ordered/Subset

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

A subset $T \subseteq S$ is said to be close packed in $\left({S, \preceq}\right)$ iff:
 * $\forall a, b \in S: a \prec b \implies \exists c \in T: a \prec c \prec b$

Beware
An intuitive understanding of this concept is likely to lead you astray.

Note that, for example, the closed real interval $\left[{0 \,.\,.\, 1}\right]$ is not close packed in $\R$.

This is because, while $\R$ is close packed in itself, and so is $\left[{0 \,.\,.\, 1}\right]$, the elements $2$ and $3$ in $\R$ have no elements of $\left[{0 \,.\,.\, 1}\right]$ between them.

This is just how the definition is constructed.

Also known as
The term close-packed is used interchangeably with densely ordered.

Also see
Compare with the topological concepts:


 * Dense-in-itself
 * Everywhere Dense
 * Nowhere Dense