= Supremum of an open disc under product and lexicographic orders
For $D=\{(x,y):x^2+y^2<1\}$, the <product order> gives <least upper bound> $(1,1)$, because each coordinate has <supremum> one. The <lexicographic order> instead has no <least upper bound>. Every point $(1,v)$ is an <upper bound>, but $(1,v-1)$ is a strictly smaller <upper bound>; a point with first coordinate below one cannot bound the disc. The same geometric <set> therefore has different order-completeness behavior under the two <partial orders>. Neither conclusion is determined by the usual drawing alone.
\Image[/past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-open-disc-orders.png]
{title=The open disc has different upper-bound geometry under two orders}
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