Dense order

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In mathematics, a partial order or total order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.

Example[edit]

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. On the other hand, the ordinary ordering on the integers is not dense.

Uniqueness[edit]

Georg Cantor proved that every two densely totally ordered countable sets without lower or upper bounds are order-isomorphic.[1] In particular, there exists an isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proof of this result uses the back-and-forth method.[2]

Minkowski's question mark function can be used to determine the order isomorphisms between the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals.

Generalizations[edit]

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related.[citation needed] Formally:

 \forall x\  \forall y\ xRy\Rightarrow (\exists z\ xRz \land zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

See also[edit]

References[edit]

  1. ^ Roitman, Judith (1990), "Theorem 27, p. 123", Introduction to Modern Set Theory, Pure and Applied Mathematics 8, John Wiley & Sons, ISBN 9780471635192 .
  2. ^ Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN 9781461488545 .

Additional reading[edit]