dense total order

A total order $(S,<)$ is dense if whenever $x<z$ in $S$ , there exists at least one element $y$ of $S$ such that $x<y<z$ . 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 $x<z$ , there exists some $y\in T$ such that $x<y<z$ . 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 $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$ .

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.

Title	dense total order
Canonical name	DenseTotalOrder
Date of creation	2013-03-22 16:40:48
Last modified on	2013-03-22 16:40:48
Owner	mps (409)
Last modified by	mps (409)
Numerical id	8
Author	mps (409)
Entry type	Definition
Classification	msc 06A05
Synonym	dense linear order
Related topic	LinearContinuum
Defines	dense
Defines	dense in
Defines	dense in itself