Pi-normality in Topological Spaces and its Generalization

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.