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Two new forms of ordered soft separation axioms

Al-shami Tareq M., El-Shafei Mohammed E.

Demonstratio Mathematica

2020

Vol. 53, nr 1

8--26

The goal of this work is to introduce and study two new types of ordered soft separation axioms, namely soft Ti-ordered and strong soft Ti-ordered spaces (i = 0, 1, 2, 3, 4). These two types are formulated with respect to the ordinary points and the distinction between them is attributed to the nature of the monotone neighborhoods. We provide several examples to elucidate the relationships among these concepts and to show the relationships associate them with their parametric topological ordered spaces and p-soft Ti-ordered spaces. Some open problems on the relationships between strong soft Ti-ordered and soft Ti-ordered spaces (i = 2, 3, 4) are posed. Also, we prove some significant results which associate both types of the introduced ordered axioms with some notions such as finite product soft spaces, soft topological and soft hereditary properties. Furthermore, we describe the shape of increasing (decreasing) soft closed and open subsets of soft regularly ordered spaces; and demonstrate that a condition of strong soft regularly ordered is sufficient for the equivalence between p-soft T1-ordered and strong soft T1-ordered spaces. Finally, we establish a number of findings that associate soft compactness with some ordered soft separation axioms initiated in this work.

Kadison’s antilattice theorem for a synaptic algebra

Foulis D. J., Pulmannová S.

Demonstratio Mathematica

2018

Vol. 51, nr 1

1--7

We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor if and only if A is an antilattice. We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.

On Weil homomorphism in locally free sheaves over structured spaces

Falkiewicz E., Sasin W.

Demonstratio Mathematica

2017

Vol. 50, nr 1

28--41

Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of Φ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.

On connected functions in ordered spaces

Jędrzejewski J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

51--58

We consider some properties of functions defined in a topological space X with values in a topological space Y. The definitions 1, 2 and 3 define the same class of functions when X and Y are equal to R with natural topology. In this article we discuss some properties of those classes and give some sufficient conditions for the space X in which real functions defined in X form the same class.