I= WEAK FORMS OF CONTINUITY, COMPACTNESS AND CONNECTEDNESS.
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Title of Thesis
WEAK FORMS OF CONTINUITY, COMPACTNESS AND CONNECTEDNESS.

Author(s)
Moiz-Ud-Din Khan
Institute/University/Department Details
Bahauddin Zakariya University, Multan
Session
1997
Subject
Mathematics
Number of Pages
101
Keywords (Extracted from title, table of contents and abstract of thesis)
continuity, compactness, connectedness, semi-open sets, semi-boundary, topological spaces, s-closed spaces, s-open functions, s-continuous functions, p-regular spaces, s-regular spaces, regular spaces

Abstract
The study of semi-open sets and their properties was initiated by N. Levine [39] in 1963. The introduction of semi-open sets raised many basic general topological questions, which has thus far led to a productive study in which many new mathematical tools have been added to the general topology tool box, many new notions have been studied. The purpose of this thesis is to study these new notions. We divide the work into four chapters.

In chapter 1, we define semi-boundary of a subset A in a space X and study its properties. Moreover, we characterize semi-continuous and semi-open mappings in terms of semi-derived and semi-boundary.

In Chapter 2, as a generalization of s-closed spaces due to G. Di Maio and T. Noiri [17], the notion of locally s-closed spaces has been introduce and investigated. In 1976 Thompson [67] introduced the notion of S-closed spaces by using the semi-open sets. In [49], Noiri introduced and investigated S-closed subspaces and locally S-closed spaces. The class of S-closed spaces is contained in the class of S-closed spaces. Noiri [17] further showed that cd-compactness due to Carnahan, weak RS-compactness due to Hong and s-closedness are all equivalent.

Sections 1 and 2 contain the brief introduction and definitions which are building blocks of notions given in the squel.

In section 3, we define and study s-closed subspaces. We also investigate some properties of sets s-closed relative to a topological space.

Noiri [49] generalized S-closed spaces due to Thompson [67] and introduced and investigated the notion of locally spaces. In section 4, we introduce a new class of spaces called locally s-closed spaces and investigate its fundamental properties. We also find some characterizations of locally s-closed spaces in Hausedorff spaces.

In section 5, we obtain some preservation theorvation theorems by using quasi-irresolute functions and semi O-closed functions. Semi O-perfect function has also been defined and its characteristics have been found. It is observed that the inverse image of s-closed set relative to a space is also s-closed relative to a space.

In section 1 of chapter 3, the brief introduction is given. In section 2, we define a new class of topological spaces called s*-regular space. This class is contained in the class of almost-regular spaces defined by Singal and Arya [63]. Several characterizations and properties of s*-regular spaces have been explored in this section.

In section 3, PΣ and weakly PΣ spaces have been studied. It is also pointed out that PΣ spaces are weaker than regular spaces and weakly-PΣ spaces are weaker than that of PΣ-spaces. We further investigate the properties of PΣ spaces and weakly-PΣ spaces. Pre-almost open, pre almost-closed and regular open functions are also defined and studied to obtain several properties of P-Σ and spaces.

In section 4, we define the notion of locally s-regular spaces. It is pointed out that the class of s-regular spaces is proper subclass of locally s-regular spaces. Some interesting results have also been proved by using weak and strong forms of continuity. In section 5, we define the strong form of regularity called P-regularity which implies semi-regularity, due to C. Dorsett [22] as well as almost-regularity due to Singal and Arya [63]. Strong s-continuous functions have also been defined and studied in this section.

In Chapter 4, we study several weak forms of continuity. In 1963, Levine [39] defined the notion of a semi-continuous function. Since then, this notion has been extensively investigated. Cameron and Woods [12] and Abd El-Monsef et.al [1] have independently defined s-continuous and strongly semi-continuous functions respectively. We introduce and investigate the notion of almost s-continuous functions. It is shown that almost s-continuity is weaker than that of s-continuity. It is also shown that almost s-continuous functions have certain similar properties to those of strong O-continuous functions obtained by Long and Herringron [40]. Although, it is shown in remark 4.4.3 that almost s-continuity and strong O-continuity are independent of each other. In section 4, we investigate the relationship among almost s-continuous functions and several modifications of continuous functions. It is shown that a surjective almost s-continuous image of a connected space is semi-connected.

In section 5, we define s-closed functions and study the properties and characterizations of s-open and s-closed functions, s-open functions were defined by D. E. Cameron and G. Woods [12].

Download Full Thesis
1558.21 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
85.58 KB
2 1 Semi-Boundary In Topological Spaces 1
146.86 KB
  1.1 Introduction 1
  1.2 Semi-Boundary 2
  1.3 Semi-Boundary And Some Functions 7
3 2 S-Closed And Locally S-Closed Spaces 10
412.33 KB
  2.1 Introduction 10
  2.2 Definitions 11
  2.3 Sets S-Closed Relative To Space 11
  2.4 Locally S-Closed Spaces 19
  2.5 Locally S-Closed Space And Some Functions 24
4 3 Weak And Strong Regularities 35
564.38 KB
  3.1 Introduction 35
  3.2 S*-Regular Spaces 35
  3.3 Pƒ And Weakly- Pƒ Spaces 42
  3.4 Locally S-Regular Spaces 52
  3.5 P-Regular Spaces 61
5 4 Some Weak Forms Of Continuity In Topological Spaces 70
368.81 KB
  4.1 Introduction 70
  4.2 Almost S-Continuous Functions 73
  4.3 Some Properties Of Almost S-Continuous Functions 76
  4.4 Comparisons 82
  4.5 S-Open Functions 89