View source for Continuous set

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A (totally) [[Ordered set|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.



====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.