I read the following excerpt from nLab but I need further explanation:
The problem Scott solved is to faithfully model untyped lambda calculus; in categorical terms, the basic problem is to construct a cartesian closed category with just one object $D$ (or rather, two objects: $D$ and a terminal object $1$), so that $D$ is closed under formation of products and function spaces: $D \cong D \times D$ and $D \cong D \Rightarrow D$. Notice that in the category of sets, the only solution is to take $D \cong 1$, so that all terms are then equal (“algorithmic inconsistency”). This is not a faithful modeling of untyped lambda calculus, which has provably unequal terms.
In 1969, Dana Scott solved this problem topologically: the terms were interpreted as continuous functions on a suitable space $D$ isomorphic to its own function space. This $D$ is called a domain. Decades later, we now know many techniques for constructing such domains as suitable objects in cartesian closed categories, but Scott’s basic insight, that computability could be interpreted as continuity, continues to exert a decisive influence today.
Exactly, how does continuity resolve the issue that the category of $Set$ cannot solve? Namely, by Cantor's theorem a set cannot be equal to its power set. But I don't understand why continuity can bypass this issue... thanks.