Definition:Close Packed/Subset

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Definition

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


A subset $T \subseteq S$ is said to be close packed in $\left({S, \preceq}\right)$ if and only if:

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


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: