الفهرس 
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Abstract
SUMMARY
Since Zadeh has introduced the concept of fuzzy sets, the
study of fuzzy sets and related problems has become a special
branch in pure and applied mathematics. tn 1968, Chang has found
that the concept of fuzzy sets provides a natural frame work for
generalizing many concepts of general topology to what he called
fuzzy topology. tn his paper [3] Chang introduced the notion of
fuzzy topological space. Subsequently. the derived fuzzy topological
notions such as closed fuzzy sets, interior and closure of a
fuzzy set ... etc .• were defined. Chang was followed by other
mathematicians such as Wong [32. 33]. Lowen [ 21, 23] and others in
studying fuzzy topological spaces. At the present time there is a
great deal of activities in the area of fUZZy topological spaces.
According to the one-one corespondence between the family of
characteristic functions on a set X and the power set of X, we
identify each subset of X with its characteristic function.
consequently an ordinary subset (resp an ordinary topological
space) is a special type of fuzzy sets (resp. Fuzzy topological
spaces) (see Definitions 1.1.1 and 1.2.1). Therefore. one would
expect some deviations of fuzzy topology from ordinary topology.
As a continuation to the study of fuzzy topology initiated by
Chang. this thesis is devoted to study some concepts in fuzzy
topology, Viz.. separation propertiesJ and fuzzy supratopological
spaces. Also. we introduce a new notions of Neighbourhoods,
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interior and closure operators, by using the new relation” )}”.
We divided this thesis into the following three chapters:
In chapter I, we give the basic concepts of fuzzy sets and
fuzzy topological spaces.
In Section 1.1, we state the basic concepts and properties of
fuzzy sets, which we use in the thesis.
In Section 1.2., we give a brief summary of all fuzzy
topological concepts necessary in the sequal. This section contains,
as far as we know, many new results which we have obtained
independently, Viz., Proposition 1.2.21., Lemma 1.2.27., 1.2.28
and Theorem 1.2.29.
In Section 1.3., we give a brief summary of the F-continuous,
open, closed and homeomorphism maps.
In Section 1.4., we give a survey of the relative fuzzy
topology given by Ghanim.
In Section 1.5., we give a survey of five methods of
generating fuzzy topology and give some examples.
In Section 1.6., we give a survey of fuzzy separation axioms.
This section contains, as far as we know, many new results which
we have obtained independently, Viz., Propositions 1.6.8, 1.6.9,
1.6.10, 1.6.12, 1.6.13, 1.6.14, 1.6.15, 1.6.16, 1.6.17 and 1.6.18.
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In Section 1.7., we give a survey of P-compactness given by
Kandil and we obtain a ne~ result given in Proposition 1.7.7.
In Chapter II, we introduced a new definition of neighbourhood
by means of the binary relation ”»”and we use it to define
interior and clasure operators of a fUZZy topological space. Ve
divided this chapter into two sections :
In section 2.1., we introduce the binary relation ”»” and we
deduce many new results, viz., Lemmas 2.1.3, 2.1.4., 2.1.9,
2.1.10. , Propositions 2.1.5., 2.1.6 .. 2.1.7., 2.1.B.and 2.1.12.
In section 2.2., we introduce some new separation properties
by using the binary relation ”»”.
Chapter III, is a continuation of the study of fuzzy supratopological
spaces initiated by nashhour et. al (19]. We divide
this chapter into two sections.
In section 3.1., we have used the definition of pre-open,
p-open, a-open and semi-open fuzzy sets given in [19] to obtain
some new results, viz Theorems 3.1.2., 3.1.3., 3.1.4., 3.1.6.,
3.1.14. and 3.1.18.
In section 3.2., we give a survey of fuzzy supratopological
spaces given in [2]. Two new results appeared in this section in
Theorem 3.2.4. and Proposition 3.2.5.