Executive summary of the project
Minor Research Project in Mathematics
MRP(S) -0453/13-14/KAKA028/UGC-SWRO
Some Studies in point set topology – some more
properties of generalized closed sets and its allied
forms
Principal Investigator
PROF. M.B. ROTTI
Topology is an indispensable of study in Mathematics with open sets and closed sets
as most important concepts in topological spaces. In the course of development of
the study of topology, open sets have been generalized by several Mathematicians.
In view of this Levin 1963, Mashhour et.al. 1982, Njastad 1965 and Abel El-Mansef
et.al 1983 introduced semi-open sets, pre open sets, -sets and -open sets
respectively. Now Njastad’s -sets are calling an -open in the literature and also
D, Andrijevic 1986 called -open sets an semipreopen sets. Since then many authors
have studied various concepts with respective to these sets in the literature.
In 1970, M. Levine initiated a study of generalized closed sets in a topological
space and studied their most fundamental properties. Generalized closed sets, their
generalizations and analogous related concepts have been studied by various
authors. P. Bhattacharya et.al.1987, S.P. Arys et.al. 1990, H.Maki et.al1984 and T.
Noiri et.al 1998 have introduced and studied sg-closed sets, gs-closed sets agclosed sets and gp-closed sets respectively. Dontchev 1995 introduced gsp-closed
sets in the literature.
With respect to neighbourhoods via g-closed sets and g-open sets are studied.
Also some generalized non-continuous functions, generalised non-open functions,
generalized non-closed functions. Further in this project we define and study some
separation axiom and some regularity axiom via-allied generalised (g-closed)
closed, generalized (g-open) open sets.
The aim of this project is to study some more properties of generalized closed
sets, generalized open sets and their allied forms. With respect to separation axioms
neighbourhood, functions.
In chapter-1, we gave introduction to the thesis and we compiled all the necessary
definitions and results which are useful in writing the thesis.
In chapter 2, we defined and obtained results of g*-neighbour hood of a point,
g*-interior operator, g*-closure operator, g*-derived set, g*-border set, g*-frontier
set and g*-exterior set.
In chapter 3, we defined and obtained results of g-pre continuous functions, gsemi pre continuous functions, g-pre open functions, quasi-g-open functions, quasigp open functions, quasi-pre-open functions, g-pre closed functions, g-semi pre
closed functions.
In chapter 4, we defined and obtained result of g-Tospaces, g*-Tospaces, g- T1
spaces, g*- T1 spaces, g*- T2 spaces, g-R1 spaces, g*-R1 spaces, g-connected
spaces, g*-connected spaces, contra- g-continuous and g*-continuous functions.
Principal Investigator