Pi-normality in Topological Spaces and its Generalization. Basics, Concepts, Properties, Examples and some Results

      Аннотация: The main aim of this work is to make a comprehensive study of a weaker version of normality called ?-normality. We give a survey study of ?-closed, ?-open, pre-closed and pre-open sets, which are the keys of both ?-normality and ?-pre-normality. Some properties of these sets are given and proved. ? -Normality is both a topological and an additive property, but neither a productive nor a hereditary property in general. The notion of ?-generalized closed sets is used to obtain various characterizations and preservation theorems of ?-normality. Some properties of almost regular as well as almost completely regular spaces are presented, and a few results of them are improved. Some relationships between ?-normality and both almost regularity and almost complete regularity are given. The important results are about presenting some counterexamples; the first one is about a semi-normal Hausdorff space but not ?-normal. The second one is about an almost normal Tychonoff space but not quasi-normal and the third one is about an almost normal Tychonoff space but not ?-normal. Some other results are presented in this work.

Ключевые слова: General Topology, Analysis, geometry, topology, pure mathematics