On topologies generated by some operators

The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.     

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).     

Some recurrence relations in finite topologies

In a number of papers (see, e.g., RZhMat, 1977, 11B586) there is given for the number T_o(n) of labeled topologies on n points satisfying the T_o separation axiom the formula $$T_0 (n) = \sum {\frac{{n!}}{{p_1 ! \ldots p_k !}}V(p_1 , \ldots p_k ),} $$ where the summation extends over all ordered sets (p₁,...,P_k) of natural numbers such that p₁+...+P_k=n. In the present paper there is found a relation for calculating, whenn?2, the sum of all terms in this formula for which p₂=1 in terms of the values V(q₁,...,q_t) withq₁+...+q_t?n-2. This permits the determination (with the aid of a computer) of the new value T_o(12)=414 864 951 055 853 499     

Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers

In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. His mathematical model is based on a special indefinite inner product of index one and its associated group of orthogonal transformations (the Lorentz group). Also, in the same monograph, the Hawking, King and McCarthy’s topology [6] is presented. This topology is physically well motivated and has the remarkable property that its homeomorphism group is essentially just the Lorentz group. Starting from the remark that the inner product and the topology above can be generated by the so-called hyperbolic complex numbers, in this paper we introduce and study two-dimensional geometries and physics generated in a similar manner, by the more general so-called complex-type numbers, i.e. of the typez=x+qy,q ? ?, whereq ²=A+q(2B), A,A, B ? ? fixed.     

Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .

     

On some relations between ideals of nowhere dense sets in topologies on positive integers

Periodica Mathematica Hungarica - We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division...     

On linear relations generated by an integro-differential equation with Nevanlinna measure in the infinite-dimensional case

A system of integral equations that can be reduced to an integro-differential equation with Nevanlinna measure is considered. The families of maximal and minimal linear relations are defined and their holomorphy is established. It is proved that the operators inverse to continuously invertible restrictions of the maximal relations are integral.     

On ordinal-metric intersection topologies

We consider the intersection topology formed by the ordinal topology on ω₁ and a separable metric topology. Such a space is never perfectly normal, but it can be normal or perfect; which one depends on the model of set theory and the choice of the metric topology.     

On similarity between topologies

Let T ₁ and T ₂ be topologies defined on the same set X and let us say that (X, T ₁) and (X, T ₂) are similar if the families of sets which have nonempty interior with respect to T ₁ and T ₂ coincide. The aim of the paper is to study how similar topologies are related with each other.     

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.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Linear relations generated by an integral equation with Nevanlinna operator measure

We define families of maximal and minimal linear relations generated by an integral equation with Nevanlinna operator measure and prove their holomorphic property. We also prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We apply the obtained results for proving the constancy of deficiency indices of some integral and differential equations.     

New convergence results for continued fractions generated by four-term recurrence relations

In the study of simultaneous rational approximation of functions using rational functions with a common denominator (which can be viewed as the ‘German polynomial’ problem in simultaneous Pade´approximation, cf. [1]) the quest for convergence results lead to the study of generalized continued fractions, a type of Jacobi—Perron algorithm [2,3]. It then becomes important to exploit the connection between the convergence of the generalized continued fraction and the solutions of the associated difference equation (cf. [4,5]).