# Continuous set

A totally ordered set $X$ all proper cuts of which are Dedekind cuts, i.e. in any partition of $X$ into two non-empty subsets $X_1$ and $X_2$ such that every element of $X_1$ precedes every element of $X_2$, either $X_1$ has a greatest element but $X_2$ no smallest element, or $X_1$ has no greatest element but $X_2$ has a smallest element.

Thus, a continuous set is a conditionally-complete lattice which is a dense total order. The phrase "continuous set" is not used in the Western literature.

How to Cite This Entry:
Continuous set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Continuous_set&oldid=33796
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article