Extension structures and compactifications

Basic results on compactifications are presented applying the notion of extension structure. Each extension structure has a canonical completion. The related completion constructions can be applied, for instance, for generating completion theorems in algebra, lattice theory and general topology, in particular they lead to a universal completion for Cauchy-spaces in the fuzzy filter case. Since compactifications can be identified with special Cauchy-completions, even different types of compactifications can be generated. Among others, we present new results on the Richardson compactification in the fuzzy filter case applying new results on fuzzy filters. This type of compactification was treated previously by the authors (1993).