dense total order
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 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 |
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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 |