In the cpo on the right (seen also in § [dt-0040]), the element x is not the lub of its compact approximations. The only compact element that approximates x is \bot , and the least upper bound of the set \{\bot \} is just \bot , not x. That is:
\mathop {\downarrow }(x) = \{\bot \}
but:
x \neq \bigsqcup \{\bot \}
