Erweiterungen topologischer Räume
An extension principle for \(T_ 0\)-spaces is formulated which includes several well-known compactifications. As an application we get, e.g., the following: For any non-empty, open subset \(D\subseteq {\mathbb{R}}^ n\), the ring of all \(C^{\infty}\)-functions on D satisfies the pm-condition (i.e., every prime ideal is contained in a unique maximal ideal) and induces a Stone-Čech compactification of D. Finally, it is shown that the compact \(T_ 2\)-spaces are exactly the \(T_ 1\)-retracts of Stone spaces, and the \(T_ 2\)-compactifications of a given space are characterized similarly. All these results are derived from some general facts on semirings stated in the first part of the paper.
scientific article; zbMATH DE number 3909309