countability and sequentiality in fuzzy topological spaces abstract lowen s good extension criterion is used to test countability and sequentiality definitions in fuzzy topology quasi coincidence is found to yield better extensions that classical set membership

quasi coincidence and quasi fuzzy hausdorff the concept of quasi coincidence of a fuzzy point to a fuzzy set is investigated the notion of a quasi fuzzy hausdorff topological space is introduced and its close relationship with the properties of fuzzy hausdorff and hausdorf is shown

dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

short communication characterization of fuzzy topologies by s neighbourhood systems using chang s definition of a fuzzy topological space or fts for short 1 we introduce the notion of s neighbourhoods of a fuzzy point by which we succeed to characterize a fuzzy topology this answers warren s open question 4

matching cutsets in graphs of diameter 2 we say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue and every vertex has at most one neighbour coloured in the opposite colour in this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset in particular we present a polynomial time algorithm which solves this problem for graphs of diameter two

probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

periodic solutions of a second order nonlinear system abstract we discuss the existence of periodic solutions of the system x t d dt grad f x t g t x t e t under various asymptotic behaviours of g

fuzzy closure spaces abstract fuzzy topological spaces do not constitute a natural boundary for the validity of theorems but many results can be extended to what are called fuzzy closure spaces or fcs s for short the notions of a subspace a sum and a product are extended to fcs s the hereditary additivity and productivity behaviour of compactness in fcs s is investigated and some weak forms of compactness and fuzzy continuous functions in fcs s are introduced the interaction between fuzzy proximity spaces and fcs s is investigated a necessary background is included for completeness

sequential convergence in fuzzy neighborhood spaces abstract we introduce the notions of sequential convergence and the first axiom of countability in fuzzy neighborhood spaces we show that these notions fulfil good extension criterion attributed to lowen convergent sequences and cauchy sequences are studied in the light of fuzzy probabilistic pseudometric space

a complexity dichotomy for matching cut in bipartite graphs of fixed diameter abstract in a graph a matching cut is an edge cut that is a matching matching cut is the problem of deciding whether or not a given graph has a matching cut which is known to be np complete even when restricted to bipartite graphs it has been proved that matching cut is polynomially solvable for graphs of diameter two in this paper we show that for any fixed integer d 3 matching cut is np complete in the class of graphs of diameter d this resolves an open problem posed by borowiecki and jesse jozefczyk 2008 6 we then show that for any fixed integer d 4 matching cut is np complete even when restricted to the class of bipartite graphs of diameter d complementing the hardness results we show that matching cut is polynomial time solvable in the class of bipartite graphs of diameter at most three and point out a new and simple polynomial time algorithm solving matching cut in graphs of diameter 2

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

periodic solutions for a kind of second order neutral differential systems with deviating arguments by means of the generalized continuation theorem of coincidence degree theory a kind of second order neutral differential systems with deviating arguments as followsd 2dt 2 x t cx t r ddtgradf x t gradg x email protected t p t is studied a new result on the existence of periodic solutions is obtained which relates to the deviating arguments t t and r

some new results on the existence of periodic solutions to a kind of rayleigh equation with a deviating argument abstract by employing the continuation theorem of coincidence degree theory we study a kind of rayleigh equation with a deviating argument as follows x t f x t g x t t p t some new results on the existence of periodic solutions are obtained

forty years of hutton fuzzy unit interval this note is a very short history of a specific fuzzy topological space the hutton s fuzzy unit interval 0 1 l which occupies one of the central positions in fuzzy topology on that occasion the superiority of l valued topology over 0 1 valued topology is being discussed we then show how the hutton s concept evolved after obvious identifications we can today say that 0 1 l is the tensor product 0 1 l of the lattices 0 1 and l an account of the consequences of this new view point is hoped to appear thereafter

on stable cutsets in line graphs we answer a question of brandstadt et al by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is np complete conversely the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently the proof of our np completeness result is based on a refinement on a result due to chvatal that recognizing decomposable graphs with maximum degree 4 is an np complete problem here a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v we also discuss some open problems on stable cutsets

a theorem on sequentiality and compactness with its fuzzy extension abstract an amended proof of a theorem of franklin s on sequentiality and sequential compactness is presented and a corresponding fuzzy version formulated using the theory of quasi coincidence

fuzzy topologies with specified level properties abstract recently the author and p wuyts have studied the problem of defining fuzzy topologies when one is given the family of closure operators or the family of level topologies this paper is concerned with the relationship between these two problems when the family of topologies is a chain with the lt property there is a natural family of closure operators that generates the wuyts maximum i e the largest fuzzy topology with the given level topologies as a corollary it is shown that the fuzzy neighborhood spaces are precisely the pseudo regular spaces in general the relationship between the two problems is weak for example given an lt family of topologies and an l ftp family of semi closure operators there is a fuzzy topology associated with both if and only if the 0 level closure generates the 0 level topology and the closure of a set is always closed in the level topology lastly it is shown that when a function s domain has a wuyts maximum as its topology fuzzy continuity is equivalent to level continuity

fuzzy points and local properties of fuzzy topological spaces abstract we prove that three theorems of c k wong on local properties of fuzzy topology are wrong thereby we discuss his notions of fuzzy point c 1 separability and local compactness

invariant probabilistic metrizability of fuzzy neighbourhood groups we introduce the notions of n local compactness and n openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups together with some characterizations for them in particular we prove that a fuzzy neighbourhood space is n locally compact if and only if its level spaces for all 0 1 are locally compact a similar criterion is established for n open functions we also introduce the notion of fuzzy absolute value functions on groups we show that there exists a 1 1 correspondence between invariant fuzzy probabilistic pseudo metrics on groups and fuzzy absolute value functions we establish theorems on fuzzy probabilistic pseudo metrizability of fuzzy neighbourhood groups finally fuzzy probabilistic metrizability of quotient fuzzy neighbourhood groups is also taken into account

a note on the relation between g and q theories for first countability and sequentiality short communication abstract first countability and quasi first countability are shown to coincide even without the introduction of zero valued points for the former property but fuzzy sequentiality provides an example of non coincidence of membership based theory with q theory

on separation axioms in fuzzy topological spaces fuzzy neighborhood spaces and fuzzy uniform spaces submitted by l zadeh a certain number of separation axioms for fuzzy topological spaces are provided all of which are good extensions of the topological t t or t2 all valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones in this paper we introduce and study a number of separation properties of fuzzy topological fuzzy neighborhood and fuzzy uniform spaces all of which are extensions of t t and t in topological spaces we started from the viewpoint that separation axioms at least t originated in the need to guarantee in some sense uniqueness of limits and we tried to generalize the topological t axiom in that direction it appears that a whole range of axioms from a fairly weak condition wt up to a rather strong one tj all give rise to acceptable extensions of the topological property moreover it seems likely that it will not be possible to single out one axiom in fuzzy topology as the best extension of topological t2 we showed in a preceding paper 181 on completions of fuzzy uniform spaces that tj as well as wt collapsing with wt in uniform spaces have a specific role to play in the same sense wt has already shown its usefulness in the study of hyperspaces 131 and t2 in its relation with compactness 191 as to our other axioms from t to wt they all seem to be natural generalizations of the corresponding properties in topological spaces and of the different t notions in fuzzy topological spaces they are moreover hereditary and t and ty excepted also productive as to the relations with other work in this field we first point out that our wt axiom is in fact the 1 hausforff separation property of hiihle 2 31 hutton and reilly in 4 and rodabaugh in 161 also studied separation current address is rijksuniversitair centrum 2020 antwerpen belgium

on the existence of periodic solutions to rayleigh differential equation of neutral type in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t c x t f x t g t x t t e t in the critical case c 1 by analyzing some properties of the linear difference operator a a x t x t c x t and using mawhin s continuation theorem some new results are obtained

fuzzy hausdorff topological spaces abstract we introduce the notion of a fuzzy hausdorff topological space and make a few observations to establish the appropriateness of this notion

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

a call for crispness in fuzzy set theory abstract this paper is a reflection of some rather negative experiences i have had during the preparation of lecture notes for a general course on fuzzy set theory the lack of standard definitions for fundamental concepts many fundamental concepts are not expanded fully enough and several proofs of important properties are incorrect

periodic solutions for a kind of second order differential equation with multiple deviating arguments by means of continuation theorem of coincidence degree theory some new results on the existence and nonexistence of periodic solutions for a kind of second order functional differential equation with multiple deviating arguments are obtained

a comparison of some hausdorff notions in fuzzy topological spaces abstract several different notions of the hausdorff separation property for fuzzy topological spaces have appeared in the literature this paper is a study of the relationships between 12 of these concepts

periodic solutions for some nonlinear differential equations of neutral type where f ir x ir ir is a caratheodory function f x is 2n periodic for every x ci e r and r e io 2n this problem has been studied by sadovskii in 151 by hale and mawhin in 6 in the case ial 1 to the previous one our interest is particularly motivated by the fact that represents a simple and meaningful model for a general phenomenon which often arises in the study of functional differential equations in fact as is well known in some cases a delayed differential equation can be seen as a perturbation of an ordinary differential equation in particular this occurs when the delayed term in the equation can be considered smaller than the ordinary one or dominated by it when such domination is available many good properties can be proved and several results such as those of 6 and 16 take advantage of this fact on the contrary when no domination is present the situation is much more complicated and in general very little is known in problem the delayed term can be considered dominated by the ordinary one whenever ial 1 as a matter of fact in this case one can prove the invertibility of the differential operator l x x a r ux the compactness of its inverse and even some a priori estimates for the solutions consequently one can reduce to a fixed point problem and solve it by means of some topological degree argument however when 11 1 so that the weight of the two terms x t and x t t is the same the above properties fail and moreover for some delays the kernel of the differential operator lx x a r is infinite dimensional therefore one must look for different methods problem is thus a simple but nevertheless significant example of the considerations illustrated by hale in 5 about the difficulty of undertaking a unified approach in the study of functional differential equations at the same time it provides an analogy between neutral equations and the wave equation in the line suggested by hale in 5

algorithms solving the matching cut problem in a graph a matching cut is an edge cut that is a matching matching cut is the problem of deciding whether or not a given graph has a matching cut which is known to be np complete this paper provides a first branching algorithm solving matching cut in time o 2 n 2 o 1 4143 n for an n vertex input graph and shows that matching cut parameterized by the vertex cover number g can be solved by a single exponential algorithm in time 2 g o n 2 moreover the paper also gives a polynomially solvable case for matching cut which covers previous known results on graphs of maximum degree three line graphs and claw free graphs

periodic solutions for generalized high order neutral differential equation in the critical case abstract by applying mawhin s continuation theory and some new inequalities we obtain sufficient conditions for the existence of periodic solutions for a generalized high order neutral differential equation in the critical case moreover an example is given to illustrate the results

periodic solutions to a second order nonlinear neutral functional differential equation in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t cx t f x t x t g t x t t e t in the critical case c 1 by employing mawhin s continuation theorem and some analysis techniques sufficient conditions are given for the existence of periodic solutions

periodic solutions to neutral differential equation with deviating arguments by employing the continuation theorem of coincidence degree theory and some analysis techniques we study a kind of neutral differential equation with deviating arguments as follows x t cx t r f x t g x t t p t some new results on the existence of periodic solutions are obtained our result generalizes main work of wang app math lett 12 1999 41 44

on connections between the countability properties of x ω t and x t abstract the connections between some countability properties of a topological space x t and its generated fuzzy topological space x t are investigated

fuzzy topology with respect to continuous lattices abstract fuzzy topology with respect to a continuous lattice generalises the classical theory continuous functions from a topological space to a lattice with the scott topology constitute a fuzzy topology corresponding to that of lower semi continuous functions with respect to the closed unit interval fuzzy set membership is conveniently represented by the way below relation

on the existence of periodic solutions for neutral functional differential equation abstract by means of mawhin s continuation theorem existence criteria are established for the periodic solutions of neutral functional differential equation

periodic solutions to a kind of neutral functional differential equation in the critical case in this paper we consider a kind of neutral functional differential equation as follows x t cx t gt xt t e t in the critical case c 1 by employing the topological degree theory and some analysis techniques we obtain some new results on the existence of periodic solutions

on first countable l fuzzy topological spaces abstract the fuzzy continuity of a bijective function defined on a first countable c 1 l fuzzy topological space where l is a spatial frame is characterized by using the sequences of l fuzzy points moreover a local characterization of fuzzy continuity is given

graded sets and points a stratified approach to fuzzy sets and points abstract in this paper we define the concepts of graded set and graded point as well as the usual theoretical set operations and relations and the extension of mappings between graded sets these definitions provide a method to extend concepts of classical set theory this method is related to fuzzy set theory and generalizes some properties of the family of cuts moreover the graded points verify the following two properties which cannot happen with fuzzy points

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

on a topological model for the category of fuzzy topological spaces i we study global properties of fuzzy topological spaces which depend only on open sets fuzzy continuous functions and the relations we obtain many known theorems as purely topological theorems

frame fuzzy points and membership for a frame l the collection of l fuzzy subsets of a set x forms a locale the localic definition of points gives rise to l fuzzy points x where x x and is a prime element of l membership is defined in terms of a satisfactory local theory of l fuzzy topology emerges from this supplemented for some purposes by continuity of l

fuzzy topological spaces we start by discussing a class of special fuzzy topological spaces that is the product induced spaces s first we show that every fuzzy topological space is topologically isomorphic with a certain topological space and then proceed to prove every open fuzzy set is defined by some lower semicontinuous function taking this as the background we introduce the concept of dual points 6 and thus establish a kind of neighborhood structure of fuzzy points such that the q neighborhood 4 which is one of the important notions in fuzzy topology and the neighborhood arc integrated in this structure this neighborhood structure will be the core of our developing the theory of fuzzy topological spaces we introduce the concept of strong quasi discoincident 1 i and so give a group of fuzzy separation properties which is a most natural generalization of the usual separation properties next we introduce a kind of fuzzy metrics and use this metrics directly to discuss the fuzzy metric space by means of fuzzy points wc define kinds of uniformities discuss their fundamental properties and extend weil s theorem on usual topology to fuzzy topological spaces and hence obtain their separation character naturally these fuzzy uniformities can still be characterized by a family of fuzzy metrics finally we discuss the problem of fuzzy mctrization on fuzzy topological spaces and obtain a fuzzy metrization theorem which contains the nagata smirnov theorem as a special example

a categorical accommodation of various notions of fuzzy topology different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology to remedy this a new fuzzy topological category fuzz is defined it is a significant generalization of previous definitions it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed by identifying each with a different subcategory of fuzz it therefore allows us to conveniently analyze the relative merits of previous definitions give coherence to known results and indicate appropriate directions for future development and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness