الفهرس 
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Abstract
Fuzzy sets were created by L. A. Zadeh [84] in 1965 in order
to be able to obtain a more distinctive description of some
phenomena than the one which is offered by systems based on
classical two-valued logic and classical set theory. Since that
time fuzzy set theory is infiltrating in almost all mathematical
theories and applied mathematical techniques that are based on
classical black-or white set theory.
Because the concepts of fuzzy subsets correspond to physical
situation in which there is no precisely defined criterion for
membership, fuzzy subsets have useful and increasing applications
in various fields including probability theory and information
theory. Thus developments in abstract mathematics using the idea of
fuzzy sets posses sound footing.
In 1.968 Chang [15] wrote his paper on fuzzy topological spaces
based on a straightforward generalization of union and intersection
to fuzzy sets. Since, then a lot of contributions to the
development of fuzzy topology have been published. Important
efforts were made on the fuzzification of local notions. Lots of
attention has been paid to the linking of ordinary topological
spaces and fuzzy topological spaces leading to Weiss’s [77] concept
of induced fuzzy topological spaces and the important operations w
and L introduced by Lowen [57]. In 1990, 1991, Prade Vicente and Macho Stadler [73],
introduced the notion of strong separation axioms in fuzzy
topological spaces which are a generalization of the separation
axioms due to (Hutton, B. and Reilly [31]) and (Lowen, R. and
Wuyts, P. [81]). Also, they introduced the notion of strong
continuous (resp. open) mappings.
In [1] the author follows the same technique in [73] and
introduced the strong To, strong R, and strong R1 axioms, but by
using the operation method [5].
Dimension theory is a branch of topology devoted to the
definition and study the notion of dimension in certain clases of
topological spaces.
Dimension theory of bitopological spaces has been
independently studied by Dusan [16] and Dvalishvili [18].
The present thesis consists of six chapters.
Chapter 0 is devoted to give an exposition of a some needed
definitions and preliminaries to be used throught this thesis.
In Chapter I, it is introduced and studied further properties
of fuzzy supratopological spaces such as, separation axioms, FSTi’
(i = 0,1,2,2,3,3,4), the strong from of these separations S.FSTj
and the strong fuzzy supracontinuous (resp. strong fuzzy supraopen,
strong fuzzy supraclosed) mappings which generalize the notions, fuzzy
supracontinuous (resp. fuzzy supraopen, fuzzy supraclosed)
mappings in [7]. Also, a fuzzy supratopology is generate from a
fuzzy bitopological space (x,T1, T2) by using the operator e12: Ix~r
defined by eI2 (J1.) =71• cl (J1.) n T2• cl (J1.).
The operator e12 induces a fuzzy supratopology Ts and the space
(x,Ts) is called a fuzzy supratopological space. Moreover, for these
spaces it is introduced the notion of continuous (resp. open,
closed) mappings. These notions generalize the notions of fuzzy
pairwise continuous (resp.fuzzy pairwise open, fuzzy pairwise
closed) mappings between fuzzy bitopological spaces.
The results of section 1.4 have been accepted for publication
in [43].
In the first section of Chapter II. we generalize the notions of 01.-
continuous (resp. strong continuous) and C1..openCrespo strong open)
due to Prade [72] to fuzzy bitopological spaces. These notions are
generalizations of the concepts of fuzzy pairwise continuous (resp.
fuzzy pairwise open, fuzzy pairwise closed) mappings.
The results of this section are submitted for publication in
[46] •
In the second section of Chapter II we introduce separation
axioms on fuzzy supra topological spaces that induced from fuzzy
bitopological spaces, then we compare them with the separation
axioms given in [38].
The third Section is concerned with strong fuzzy separation
axioms in fuzzy bitopological spaces. We generalize the separation vaxioms
given in [1], [73], to fuzzy bitopological spaces. These
separation axioms are generalizations of axioms given in [11] and
[38] .
The resul ts of the second section have been accepted for
publication [43], and the results of the third section are sumitted
for publication, [44].
Chapter III is concerned with the notions, FS-Q,,-compact, FP-Q,,-
compact, FP”-a- compact, FP”-Q,,-compact and FP”-compact for fuzzy
bitopological spaces. Also these notions are introduced for compact
fuzzy subsets. It is proved that the concept of FS-Q,,-compact is
productive. One can construct the one-point compactification by
using this notion.
The notions Fp”-a-compact, FP”-Q,,-compactand FP”-compact are
defined for fuzzy supratopological spaces that generated from fuzzy
bitopological s~aces. The first one of them is defined by using the
a-shading, the second by the relation of q-neighbourhood and the
third one is defined by the point cover.
Also I would like to attention that the results of section 3.4
have been accepted for publication in [43].
Chapter rv, deals with the fuzzy supra proximity spaces and
fuzzy supra proximity spaces that induced by fuzzy biproximity
spaces. It is proved that every fuzzy supra proximity corresponds
a fuzzy supratopology. Alsol It is proved that every fuzzy visupratopological
space admits a fuzzy supra proximity iff the space
is FSRz(l/2).
In section 4.2 we prove that the two notions of induced fuzzy
supra proximity and induced fuzzy supratopology are compatible.
Also, one can prove that a fuzzy supra topological space is
fuzzy supra proximable iff its a-level spaces are supra proximable
for each aE{O,l).
In section 4.3, we investigate certain properties of a set X
equipped with two fuzzy proximities 61 and Oz (or fuzzy quasiproximities O2 = °
), such space {X,0]1 c32} is called fuzzy
biproxiimity space. We use the fuzzy proximities 61 and O2 to
generate a relation fJs which is a fuzzy supra proximity on X. We
state and prove’some results on the space (X,fJs). The main result
is: if the fuzzy bitopological space (X,711 72) is FP*-compact and
FP*T2, then the associated fuzzy supra topolgical space (X,7
C12
) has
a unique compatible fuzzy supra proximity.
All results in this chapter have been accepted for publication
in [45].
Chapter (V) I contains the defnition and properties for the
dimension theory of parwise comletely regular spaces. It is
introduced the notion of the large inductive dimension pfx.Ind and
study some of its properties. The main result is that: If the
bitopological space (X,T1I Tz) is pairwise normal, then viipfx
·Ind X=p.Ind X, where p.Ind is the dimension functions defined
in [18]. Also, it is introduced the dimension function p.dim X by
using the notion of pairwise upper and pairwise lower zero sets
[24] and study a characterization of the function dim in terms of
pairwise upper and pairwise lower co-zero-sets.
The results of this chapter has been accepted for publication
in [41].
I would like to point out that, one of aims of this thesis
was, generalization the results of chapter V to fuzzy bitopological
spaces. There are some problems related to the notion of the
”boundary” and the notion of ”the cover which are basic notions for
defining the dimension Ind and dim, respectively.
Also the boundary and the’ covetn6tions have not taken their final
form. In the future, we hope to generalize these notions for fuzzy
bitopological spaces.