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

A subset of a total order is dense in if for every such that , there exists some such that . Because of this, a dense total order 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 . On the other hand, the rationals are dense, since whenever and are rational numbers, it follows that is a rational number strictly between and . Also, both and the irrationals are dense in .

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.