Definition:Densely Ordered/Warning

Definition
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.