I know that the lexicographic order on $\mathbb{C}$ turns it into an ordered set. I just wonder if under this ordering, $\mathbb{C}$ has the least upper bound property
2025-01-12 19:14:31.1736709271
on the lexicographic order on $\mathbb{C}$
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No. The set $$\{ z \in \mathbb{C} : \Re z < 0 \}$$ is bounded above by $1$, and has no supremum.