Basic Theorems of the Theory of Pre-ends for Space Domains

In the article, we present two constructions based on the theory of pre-ends for domains in the Euclidean space $\mathbb R^n$. The former, using factorization of a set, allows us to compactify every domain in $\mathbb R^n$ homeomorphic to a ball. Applying the latter, we obtain the Hausdorff completion of a domain. As a result, we establish some analogs of the Carathéodory theorem on the boundary behavior of quasi-isometries.