Remarks on enlargements of generalized topologies

The concept of enlargement of a generalized topology was introduced by á. Császár [4]. He also introduced the concepts of (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity on enlargements. In this paper, we characterize the (κ,λ)-continuity and introduce the concept of strong (κ,λ)-continuity on enlargements. In particular, we study characterizations for the strong (κ,λ)-continuity and the relationships among (μ,ν)-continuity, (κ,λ)-continuity, strong (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity.     

Normal generalized topologies

The concept of normality is defined for generalized topologies in the sense of [1], a few properties of normal spaces are proved, and their characterization with the help of a suitable form of Urysohn’s lemma is discussed. Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.     

Remarks on products of generalized topologies

We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies, which shows that the definition of product of generalized topologies is quite reasonable.     

Lattice of generalized topologies

We prove that the family of all generalized topologies on a nonempty set is a lattice, neither distributive nor complemented. We define the direct sum of two generalized topologies and characterize the direct sum.     

Product of generalized topologies

The definition of the product of topologies is generalized in such a way that topologies are replaced by generalized topologies in the sense of [3].     

More remarks on Souslin properties and tree topologies

We investigate separation properties of ω₁-trees. We show that the property γ of Devlin and Shelah is equivalent to hereditary collectionwise normality. We show that monotone normality and divisibility are both equivalent to orderability. Finally we show that Souslin trees are examples of trees with property γ which are not retractable.     

Enlargements and generalized topologies

The paper discusses the generalization of a construction described in [6] for the case when the starting point topology is replaced by a generalized topology. Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.     

Monotonicity properties of operations on generalized topologies

Summary In the paper [5], several operations on generalized topologies are considered. They are not monotone in general, but an old result on monotonicity may be sharpened.     

Some remarks on generalized roundness

By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank  ≥ 2 equals 1. This answers a question of J-F. Lafont and S. Prassidis.      

Two remarks on generalized hexagons

In this paper, the point-distance-2-regular hexagons are characterized as the only hexagons for which the intersection sets have size one and containing on ovoidal subspace all the points of which are 3-regular. We also give a characterization of the finite split Cayley hexagon of even order.     

On extremally disconnected generalized topologies

Extremal disconnectedness is further investigated for generalized topological spaces. It is found that extremally disconnected generalized topological spaces are a rich source of generalized lower semi-continuous and generalized upper semi-continuous mappings.     

Ultratopologies generated by generalized topologies

Summary It is proved a rather general version of the statement that if the union of arbitrary elements of a system λ always belongs to λ then the intersections of elements of λ constitute an ultratopology (i.e. a topology where intersections of open sets are open).     

Separation axioms for generalized topologies

The fundamental concepts in topological spaces, in particular separation axioms, are presented in a manner that open sets are replaced by more general ones.     

Mixed constructions for generalized topologies

As generalization of the construction of I-τ where τ is a topology and I is an ideal [11], and similarly of $ \mathcal{H} $ -µ where µ is a generalized topology and $ \mathcal{H} $ is a hereditary class [5], we introduce generalized topologies starting from two generalized topologies µ and µ′.     

Further remarks on the non-degeneracy condition

The structure of the set of positive solutions of the semilinear elliptic boundary value problem depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R ², and it is either symmetric or convex.     

Generalized open sets in generalized topologies

Summary We study the even power means of a sum analogous to Dedekind sums, and give a sharp asymptotic formula.     

On the lattice of principal generalized topologies

Periodica Mathematica Hungarica - In this paper we investigate the structure of the complete lattice of principal generalized topologies, employing the notion of ultratopology. On any partially...     

Modification of generalized topologies via hereditary classes

The purpose of the paper is to show that the construction leading from a topology and an ideal of sets to another topology remains valid, together with a lot of applications, if topology is replaced by generalized topology and ideal by hereditary class.