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Dense ordered set

From Encyclopedia of Mathematics
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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://www.encyclopediaofmath.org/index.php?title=Dense_ordered_set&oldid=39616