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Abstract
Since the inception of fuzzy set theory [ see Zadeh (1965) and Goguen (1967)], the efforts of many researches have been directed
to develop fuzzy analogues of basic concepts of classical mathematics
and to work out correspoinding theories. Topological structures
on lattices of fuzzy sets were first considered in 1968 by Chang,
later Chang’s ideas were developed in essentially different directions
by [Goguen (1973), Wong (1973), Lowen (1976) and (1977),
Warren (1978) and (1979), Hutton (1980), Rodabaugh (1980) and
(1991), Kerre (1989) and (1991), etc].
It is easy to see that they have always investigated fuzzy objects
with crisp methods. Fuzziness in the concept of openness of a fuzzy
set has not been considered, which seems to be a drawback in the
process of fuzzification of the concept of topological spaces. By
the ends of eighties and beginning of nineties many mathematician
remarked that the fuzziness in these extensions is not enough, since
we handle with fuzzy subsets but the handling is crisp.
For this reason many mathematicians try to make a fuzzy treatment
for this structures, fuzzification of openness was first initiated
by [H¨ohle(1980), Kubiak (1985) and ˇSostak (1985)]. In 1991, from
a logical point of view, Ying (1991) studied H¨ohle’s topology and
called it fuzzifying topology. This fuzzy topology is an extension
of both crisp topology and Chang’s fuzzy topology and each fuzzy
subset has a degree of openness, in the sense that not only the
objects are fuzzified, but also the axiomatics, i.e., this topology
makes a fuzzy set be open to some extend, that is to say the open
property becomes fuzzy. [ˇSostak (1989a), (1989b), (1989c), (1990)
and (1996)], gave some rules and showed how such an extension
can be realized. In (1992), fuzzy topological spaces in ˇSostak sense
Typeset by AMS-TEX
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were independently redefined by [ Ramadan (1992)], under the
name of smooth topological spaces using lattices. It has been developed
in many directions [Hazra et., al., (1992), Chattopadhyay
and Samanta (1993), El-Gayyar et. al., (1994), H¨ohle and ˇSostak
(1995), Demirci (1997), Zhang (1999) and (2002), Kotze (2003),
Kubiak and ˇSostak (2004), Fang Jinming (2006)].
Attanassov (1986) introduced the idea of intuitionistic fuzzy
set. ¸Coker and coworker [(1996), (1997) ] introduced the idea of
the topology of intuitionistic fuzzy sets. Recently, Samanta and
Mondal (2002) introduced the notion of intuitionistic gradation of
openness which a generalization of both of fuzzy topological spaces
[ˇSostak (1986)] and the topology of intuitionistic fuzzy sets [ ¸Coker
and Demirci (1996), ¸Coker (1997)].
Our main object is to investigate more further the structure
of the intuitionistic supra gradation of openness and intuitionistic
gradation of openness when some information are known about
their fuzzy structure.
This thesis includes five chapters:
Chapter I, is of introductory nature, providing the reader
with results concerning, fuzzy sets, intuitionistic fuzzy sets, fuzzy
topologies, intuitionistic fuzzy topologies and intuitionistic gradation
of openness.
Chapter II, we introduce the concept of ˘C-IF closure spaces
which is a generalization of the IF-closure space introduced by [Kim
and Ko (2004)]. Also, we introduced some separation axioms in
˘C-IF closure spaces. Finally, we study the notion of IF-closure systems
and study many important properties of ˘C-IF closure spaces
and IF-closure systems.
Chapter III, we introduce and study the concept of G-closure
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operator induced by an intuitionistic fuzzy topological space in
view of the definition of Samanta and Mondal. We show that
it is an IF-closure operator. Furthermore, it induces an intuitionistic
gradation of openness which is finer than a given intuitionistic
gradation of openness. We investigate some properties
of GIF-closure operators. Also, we define (r; s)-generalized fuzzy
(semi, weakly semi) closed sets in an intuitionistic fuzzy topological
spaces. Moreover, We investigate some properties of GIFsemicontinuous
mapping.
Chapter IV, we shall give various characterizations of (r; s)-
fuzzy regularity and (r; s)-fuzzy almost regularity with the help of
quasi-neighbourhood [Pu and Liu (1980)], µ-closure and ±-closure
operators [Kim and Park (2000)]. Also, we give some properties of
(r; s)-T2 space.
Chapter V, we have used the intuitionistic supra gradation of
openness which created from an intutionistic fuzzy bitopological
spaces to introduce and study the concepts of continuity, some
kinds of separation axioms and compactness.
Most of the results of this thesis either have been accepted or
submitted for publications as follows:
(1) Intuitionistic supra gradation of openness, Applied Mathematics
and Information Sciences, Vol. 2, No. 3, (2008), 291-307.
(2) Intuitionistic fuzzy G-closure operators, International Review
of Fuzzy Mathematics (IRFM), Vol. 3, No. 1, (2008), 37-53.
(3) Several types of intuitionistic fuzzy semiclosed sets, J. Fuzzy
Mathematics (submitted)
(4) Some properties of (r; s)¡T2 spaces, Journal of the Egyptian
Mathematical Society (submitted)
(5) Characterizations of some IF separation axioms, J. Fuzzy
Mathematics (submitted)
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(6) IF-closure systems and IF-closure operators, The 22nd International
Conference on Topology and its Appications [Helwan 7-8
july (2008)], and submitted for ”Journal of the Egyptian Mathematical
Society ”