Π -Normality In Topological Spaces And Its Generalization
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Date
2013-11
Authors
Saad Thabit, Sadeq Ali
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
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.
Description
Keywords
Π , Topological Spaces