dense total order

A total orderMathworldPlanetmath (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 intervalMathworldPlanetmath has nonempty interior.

A subset T of a total order S is dense in S if for every x,zS such that x<z, there exists some yT 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 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 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.

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