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.

Comments

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.