dense total order

A total order is dense if whenever in $S$, there exists at least one element $y$ of $S$ such that . That is, each nontrivial closed interval has nonempty interior.

A subset $T$ of a total order $S$ is dense in $S$ if for every $x,z\in S$ such that , there exists some $y\in T$ such that . Because of this, a dense total order $S$ is sometimes said to be dense in itself.

For example, the integers with the usual order are not dense, since there is no integer strictly between $0$ and $1$. On the other hand, the rationals $\mathbb{Q}$ are dense, since whenever $r$ and $s$ are rational numbers, it follows that $(r+s)/2$ is a rational number strictly between $r$ and $s$.Also, both $\mathbb{Q}$ and the irrationals $ℝ\setminus \mathbb{Q}$ are dense in $ℝ$.

It is usually convenient to assume that a dense order has at least two elements. This allows one to avoid the trivial cases of the one-point order and the empty order.