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The main aim of this work is to make a comprehensive study of a weaker version of normality called pi-normality. We give a survey study of pi-closed, pi-open, pre-closed and pre-open sets, which are the keys of both pi-normality and pi-pre-normality. Some properties of these sets are given and proved. pi -Normality is both a topological and an additive property, but neither a productive nor a hereditary property in general. The notion of pi-generalized closed sets is used to obtain various characterizations and preservation theorems of pi-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 pi-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 pi-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 pi-normal. Some other results are presented in this work.

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Dr. SADEQ ALI SAAD THABIT.Studied pure mathematics, topology at Universiti Sains Malaysia. Assistant professor at Hadhramout University, Almahra, Yemen.

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