dense total order
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 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.
|Title||dense total order|
|Date of creation||2013-03-22 16:40:48|
|Last modified on||2013-03-22 16:40:48|
|Last modified by||mps (409)|
|Synonym||dense linear order|
|Defines||dense in itself|