Weak separation axioms in generalized topological spaces

We study a unified approach of weak separation axioms in generalized topological spaces.     

Separation axioms in fuzzy topological spaces

This paper provides a hierarchy of separation properties for fuzzy topological spaces which are generalizations of the standard topological notions.     

On separation axioms in fuzzy topological spaces,fuzzy neighborhood spaces,and fuzzy uniform spaces

A certain number of separation axioms for fuzzy topological spaces are provided, all of which are good extensions of the topological (T₀), (T₁), or (T₂). All valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones.     

Baireness on generalized topological spaces

Baireness on generalized topological spaces is introduced, and its characterizations and properties are investigated.     

Regular sets in generalized topological spaces

We define and study regular and locally closed sets in generalized topological spaces.     

New kinds of generalized variational-like inequality problems in topological vector spaces

In this work, we consider a generalized nonlinear variational-like inequality problem, in topological vector spaces, and, by using the KKM technique, we prove an existence theorem. Our result extends a theorem of Ahmad and Irfan [R. Ahmad, S.S. Irfan, On the generalized nonlinear variational-like inequality problems, Appl. Math. Lett. 19 (2006) 294–297].     

Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.     

Weak continuity on generalized topological spaces

We introduce the notions of weak (g, g′)-continuity and weak (ψ, ψ′)-continuity on GTS’s and GNS’s, respectively, and investigate characterizations for such functions and relationships between weak (g, g′)-continuity and weak (ψ, ψ′)-continuity.     

Contra continuity on generalized topological spaces

Contra continuous functions on generalized topological spaces are introduced, and their characterizations and properties are investigated.     

Almost continuity on generalized topological spaces

For any unbounded sequence {n _k } of positive real numbers, there exists a permutation {n _σ(k)} such that the discrepancies of {n _σ(k) x} obey the law of the iterated logarithm exactly in the same way as the uniform i.i.d. sequence {U _k }.     

Some results on generalized topological spaces and generalized systems

Summary We characterize some properties of generalized topological spaces and (g,g’)-continuity by using an interior operator defined on a generalized topological space. Also, we introduce the notions of (ψ, ψ’)-open map, gn-continuity, gn-open map and investigate their properties by using new interior (or closure) operators defined on generalized neighborhood systems of a nonempty set.     

Maximal elements in topological and generalized metric spaces

In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

Unification of generalized open sets on topological spaces

A new kind of sets called generalized μ-closed (briefly g μ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by á. Császár. The class of all g μ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine [17]) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μ _g -regular and μ _g -normal spaces have been given.     

The Hausdorff separation axiom for fuzzy topological spaces

We develop a theory of α-Hausdorff fuzzy topological spaces which is compatible with α-compactness and fuzzy continuity, and for α a certain type of member of a given lattice we obtain characterizations of the α-Hausdorff subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line. In route we give an easy proof of the Fuzzy Tychonov Theorem for α-compactness and extend the theory of one-point α-compactifications.     

Nice separation axioms

The epi-reflective hull B of an objectB in the category of locales is called anice separation axiom provided it enjoys certain properties which are natural generalizations of properties satisfied by the categories of completely regular locales and zero dimensional locales. Invariably, a family ofStone-like duality theorems ensues, distinguishing (for each large enough regular cardinal ) the full subcategory of all -LindelöfB-objects. Some corollaries for topological spaces arise, as well as some open problems, upon taking of spatial parts.This paper, consisting primarily of excerpts from the author's Ph.D. thesis, Wesleyan University, Middletown, Connecticut (1992), also represents, in part, the results of research engaged in at Charles University, Prague, during the 1992–1993 academic year, with the support of the United States Information Agency and the Fulbright Program.     

Generalized continuous functions defined by generalized open sets on generalized topological spaces

We introduce generalized continuous functions defined by generalized open (= g-α-open, g-semi-open, g-preopen, g-β-open) sets in generalized topological spaces which are generalized (g, g′)-continuous functions. We investigate characterizations and relationships among such functions.     

广义近似空间的粗糙同胚与拓扑同胚

对广义近似空间之间的映射引入并刻画了粗糙连续性和拓扑连续性,探讨了他们的性质及相互关系.证明了两个粗糙连续映射的复合还是粗糙连续的,每个粗糙连续的映射都是拓扑连续的.在此基础上引入了粗糙同胚性质和拓扑同胚性质的概念,证明了拓扑同胚性质均为粗糙同胚性质并考察了广义近似空间的诸如分离性、连通性、紧性等的粗糙同胚不变性和拓扑同胚不变性.