# Dense ordered set

From Encyclopedia of Mathematics

A linearly ordered set $(X,{<})$ with the property that if $x < y$ then there exists $z \in X$ with $x < z < y$.

Cantor showed that any countable dense unbounded linearly ordered sets are order isomorphic.

The Suslin problem asks whether a dense complete linearly ordered set without first and last elements, in which every family of non-empty disjoint intervals is countable, is order isomorphic to the set of real numbers.

#### References

- T. Jech, "Set theory. The third millennium edition, revised and expanded" Springer Monographs in Mathematics (2003). Zbl 1007.03002

**How to Cite This Entry:**

Dense ordered set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dense_ordered_set&oldid=39616