View source for Continuous set

A (totally) [[Ordered set|ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257601.png" /> all proper cuts of which are Dedekind cuts, i.e. in any partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257602.png" /> into two non-empty subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257604.png" /> such that every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257605.png" /> precedes every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257606.png" />, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257607.png" /> has a greatest element but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257608.png" /> no smallest element, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c0257609.png" /> has no greatest element but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025760/c02576010.png" /> has a smallest element.



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