Soft Topological Spaces Induced via Soft Relations

Soft relation is a basic mathematical model that can be related to several real-life data. Throughout many fields, soft relations are used to build soft topological structures. In addition, soft topological constructs are generalized methods to calculate similarity and dissimilarity of objects. Within this article, we present a new approach for directly producing a soft topology by soft relation without using base or subbase. This process is important technique for applications of soft topology. There is investigations into the relationship between soft set topologies and different relations and some of their properties are obtained.

Closure theories of powerset theories

A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using examples of powerset theories in the category Set of sets and in the category of sets with similarity relations, it is proved that these theories can be used as ground theories for closure theories of powerset theories in these two categories. Moreover, it is proved that all these closure theories of powerset theories are topological constructs. A notion of a closure operator which preserves a canonical form of fuzzy objects in these categories is introduced, and it is proved that a closure theory of a powerset theory in the ground category Set is a coreflective subcategory of the closure theory of (Zadeh’s) powerset theory, which preserves canonical forms of fuzzy sets.

The quasitopos hull of the construct of closure spaces

<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>

Distancing and Distance Devices in Childhood Schizophrenia and Borderline States: Revised Concepts and New Directions in Research

Continuing investigation of distancing and distance devices has led to elaborations of the differences between the psychotic and neurotic patients' use of distance. Investigation has led to the consideration of non-metric topological constructs to replace arithmetic ones in describing this aspect of the psychotic patient's functioning.