الفهرس 
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Abstract
The notion of a fuzzy set, introduced by Lotfi A. Zadeh ([65]) in 1965, has caused
great interest among both pure and applied mathematicians. It lias also raised
enthusiasm among some engineers, biologists, ;lhychologists, economists, and experts
in other areas, who usc (01’ at least try to use) mathclllatica.1 ideas and methods in
their research. By this notion of fuzzy set it is possible to obtain a more distinctive
description of some phenomena than the one which is offered by syst(~Il\S based
on classical two-valued logic and classical set theory. In the fuzzy set theory there
appear linguistic expressions such as ”a person is tall”. In the crisp point of view one
may say that a person is tall if his height is greater than 185 em whereas in the fuzzy
point of view this property can be determined by a function on a scale of centimetres
with values in the dosed real unit int,:-,rval [0,1]. There are many realized technical
applications showing the pradical usefulness of fuzzy theory. The first applications
of fuzzy theory appear in the late seventies. Tbey were primarily industrial such
as process control for cement kilns. In 1987 in Senrlai, Japan a fuzzy controlled
subway system started operation. Beginning with that time, fuzzy control is more
and more technically applied. In particular, a lot of consumer products, made in
Japan, is based on fuzzy technology. Examples arc t.hc following: fllz,z,ywashing
machine, fuzzy air conditioner and fuzzy vacuum cleaner. Moreover, fuzzy control is
used in camcorders, television sets, cameras (eliminating jumping of pictures), cars
(electronic fuel injection controls, automatic cruise control system), and so on.
General topology was one of the first branches of pure mathematics to which
fuzzy sets have been applied. It was in 1968, that is, three years after Zadeh’s paper
had appeared, that Chang ([6]) introduced the notion of fuzzy topology. Several
other authors continued the investigation of such spaces such as Eklund and Gahler
([11]), Gahler ([21, 24]), Geping and Lanfang ([31]), Goguen ([33f), Katsaras and
Petalas ([42]), Kerre and Ottoy ([44]) and Lowen ([47,48,49,50,51, 52,53]). In
Goguen’s paper ([33]) he considered as values of the fuzzy sets not only elements of
the closed unit interval but elements of some more general lattice L. The notion of
stratified fuzzy topology has been introduced by Lowen ([51]). By a fuzzy topology
to be stratified is meant that all constant fuzzy sets are open. Lowen defined also
the notion of induced fuzzy topology of an ordinary topology as a special stratified
fuzzy topology. Acceptable notions of fuzzy proximity were given by Artico and
Moresco ([2,3]) and by Katsaras ([40]). Some definitions of fuzzy uniform structure
neighborhood structure.
Note that R. Lowen defined in [52] a notion of fuzzy neighborhood space using
only prefilters. As a consequence, fuzzy neighborhood spaces in his sense are, up
to identifications, only special fuzzy topological spaces (d. the characterizations in
(561).
The thesis consists of three chapters and is organized as follows:
Each chapter begins with an introduction, in which are given some informations,
ideas and motivations on the contents of this chapter.
Chapter 1 is the basic chapter of the thesis which contains the definition of
the fuzzy neighborhood structure and its modification obtained by restricting to
homogeneous fuzzy filters. In this chapter it is shown that, up to an identification,
the category FNSh of homogeneous fuzzy neighborhood spaces is a full and bicorefiective
subcategory of the category FNS of fuzzy neighborhood spaces. It is also
shown that the fun category FTOP of fuzzy topological spaces is a full and bicoreflective
subcategory of FNS and that the subcategory of FTOP of all stratified fuzzy
topological spaces is a full and bicoreflective subcategory of FNSh. In Section 1.2,
we recall some of the definitions and results about fuzzy filters needing throughout
the thesis. Section 1.3 deals with the description of the first type of the fuzzy
neighborhood structure. The case of fuzzy topology is considered and it is shown
that any fuzzy topology is, up to an identification, a special fuzzy neighborhood
structure of the first type. Section 1.4 is devoted to the homogeneous the fuzzy
neighborhood structure. The fuzzy topology associated to a homogeneous fuzzy
neighborhood structure is defined analogously as the fuzzy topology associated to a
fuzzy neighborhood structure in the general sense. In this section also the stratified
fuzzy neighborhood structures are defined. Moreover, the relations between the first
type and the second type of fuzzy neighborhood structures are investigated. In case
of a fuzzy topology, considered as a special fuzzy neighborhood structure, stratified
in sense of fuzzy neighborhood structure coincides with the property of this fuzzy
topology to be stratified in the usual sense. In Section 1.5, it is shown that for any
isotone mapping h : :FLX ~ :FLX which has all properties of a fuzzy neighborhood
structure may be with except of (N2) by a ”basic construction” the greatest fuzzy
neighborhood structure less than or equal to h is constructed. Applying this basic
construction, the stratification of a fuzzy neighborhood structure h, that is, the
coarsest stratified fuzzy neighborhood structure finer than h, is introduced. In this
section also it is shown that the full subcategory of FNS of all stratified fuzzy neighborhood
spaces is a bicorcfleetive subcategory of FNS. In Section 1.6 we show,
using the ”basic construction”, that in the category FNS all initial and final lifts exists,
tha.t is, FNS is a topological category. Hence FNS is complete and cocomplete.
As a consequence, for instance, all products and sums of fuzzy neighborhood spaces
exist.
Noting that some of the results in Chapter 1 are subrnit.l.cd for publlcal.l0n III
[27l and the most of results in this chapter arc accepted for publicatioll in [2
8l-
In Chapter 2 we study fuzzy topogenous orders, in particular, fuzzy topogenous
structures and the more special fuzzy proximities in sense Katsaras ([401). In this
chapter also a modification of the notion of fuzzy proximity is considered, called
fuzzy proximity of the internal type. In Section 2.2 we show that fuzzy topogenous
structures are, up to an identification, special fuzzy neighborhood structures. More
generally, we show that fuzzy topogenous orders can be characterized by the notion
of fuzzy neighborhood prestructure which is a weakening of that of fuzzy neighborhood
structure. It is shown that up to our identifica.tiolls the ca.tegory FTS of fuzzy
topogenous spaces is a full subcategory of the category FNS of fuzzy neighborhood
spaces. In Sections 2.3 and 2.4, the notions of perfect and biperfect fuzzy top agenous
orders arc sl,ndil,d. Moreov(’r, I’lIz,z.yrd”.l.iolls aI’(’ cOlIsidcl’(,d arul it, is shown
that spedal and imp01’til,nl. typl’ of hiperkr\’ fu;t,1,y 1.0pOj!;I·1l1l1lS01’1”’1’1’ (’’’.Il Ill’ characterized
by fuzzy relations. In Section 2.5, stratified fuzzy topogenous orders and
structures are defined. It is shown that for each stratified fuzey topogcnous structure
the characterizing fuzzy neigl~borhood structure is also stratified. Here also
for fuzzy topogenous structure the related notion of stratification is introduced and
some results related to the stratification of the characterizing fuzzy neighborhood
structures are obtained. Section 2.6 is devoted to initial and final fuzzy topogenous
orders. In Section 2.7, we study fuzzy proximities. The rroperty of symmetry of a
fuzzy proximity depends on a fixed order-reversing involution of the given lattice L.
In this section a notion of symmetry of fuzzy neighborhood structure is introduced
which is defined independently on a fixed order-reversing involution ofL. In specializing
to fuzzy topogenous structures for these’ structures a new notion of symmetry
is introduced called internal symmetry. The internally symmetric fuzzy topogenous
structures give a new notion of fuzzy proximity, called fuzzy proximity of the internal
type. It is shown that a fuzzy topogenous structure is a. fuzzy proximity of the
internal type if and only if the associated fuzzy neighborhood structure is symmetric.
In Section 2.8, the notion of stratified fuzzy proximity and the relaled notion
of stratification are considered. Using this stratification of fuzzy proximities usual
proximities can be embedded canonically into the fuzzy case.
Let us here mention that some of the results in Chapter 2 are submitted for
publication in [27] and the most of results in this chapter are submitted for
publication in [29l·
Chapter 3 is devoted to a new notion of fuzzy uniform structure. In Section
3.2, we study special fuzzy filters, called relational fuzzy filte7’s and sup-fuzzy filters.
Examples of sup-fuzzy filters are the homogeneous fuzzy filters and the fuzzy
filters which only have 0 and 1 as values, called 0, l-Iuzzy filters. In Section 3.3,
by means of the relational fuzzy filters, sup-fuzzy filters. 0, I-fuzzy filters and homogeneous
fuzzy filters we define the fuzzy uniform structures, sup-fuzzy uniform
structures, 0, I-fuzzy uniform structures and homogeneo~s fuzzy uniform structures,
respectively. Fuzzy uniform structures in sense of Katsaras ([38]) are, up to identification,
0, I-fuzzy uniform structures. In Section 3.4, it is shown that the sup-fuzzy
uniform structure can be characterized by a related notion of fuzzy sytopogeno
us
structure. The 0, I-fuzzy uniform structures coincide with the fuzzy syntopogenous
structures proposed by Katsaras and Petalas in [42] in a special case. Here we
compare also our notion of fuzzy uniform st.ruc1.url’ with t.hal. of [l utf.on ([:m]). In
Section 3.5 we show that the fuzzy metrics in sense of (18] generate canonically
homogeneous fuzzy uniform structures in our sense. In Section 3.6, to each fuzzy
uniform structure U is associated a homogeneous fuzzy neighborhood structure h.
In case of a 0, l-fuzzy uniform structure and the more general case of a cascade
fuzzy uniform structure even a fuzzy neighborhood structure of the first type can be
associated, which is stratified. For the fuzzy uniform structure in sense of Katsaras,
h is, up to an identification, a fuzzy proximity of the internal type.
Noting that the most of results in Chapt.(~r :l arc 8uhlllil.kd for publicat.ion ill
[30].