In an eort to introduce a powerful invariant for C-algebras, Coward, Elliott and Ivanescu have enriched the Cuntz semigroup with a certain topological information. This topological information is coded in a binary relation << that expresses an abstract notion of compact containment. This relation has served for about forty years as a mathematical tool for creating semantic domains for interpreting logical calculi and programming languages. In Theoretical Computer Science it is known under the name of Domain Theory. It has been developed mainly for the purpose of semantics. But historically, the origins also grew out of mathematical questions. It was a surprise to discover that recently these ideas have become a tool in connection to C-algebras. In the talk, some of the historical background will be reviewed, a short introduction into the mathematical theory will be presented, the bridge to (abstract) Cuntz semigroups will be established and a result unifying results in a paper by Elliott, Robert snd Santiago on lower semicontinuous traces and quasi-traces on C-algebras will be presented.