AbstractWe use a subfamily of the Scott-closed sets of a poset to form a local completion of the poset. This is simultaneously a topological analogue of the ideal completion of a poset and a generalization of the sobrification of a topological space. After we show that our construction is the object level of a left adjoint to the forgetful functor from the category of local cpos to the category of posets and Scott-continuous maps, we use this completion to show how local domains can play a role in the study of domain-theoretic models of topological spaces. Our main result shows that any topological space that is homeomorphic to the maximal elements of a continuous poset that is weak at the top also is homeomorphic to the maximal elements of a bounded complete local domain. The advantage is that continuous maps between such spaces extend to Scott-continuous maps between the modeling local domains.