Π -Normality In Topological Spaces And Its Generalization - Thabit, Sadeq Ali Saad (2013)
The main aim of this thesis is to make a comprehensive study of a weaker version of normality called p-normality, which lies between normality and almost normality (quasi-normality). First, we give some basic definitions, properties and theorems, which we are going to use throughout the thesis. We give a survey study of Π-closed, p-open, pre-closed and pre-open sets. In particular, we study these sets in subspaces and also study the images and the inverse images of them under continuous functions. Some properties of these sets are given and proved. Π-normality is both a topological and an additive property, but neither a productive nor a hereditary property in general. The notion of Π-generalized closed sets is used to obtain various characterizations and preservation theorems of Π-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 Π-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 Π-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 Π-normal.