Structures, Operations and their Applications to Topology

A structure on a non empty set X is a collection of subsets of X. Any kind of topology on a non empty set X is a special structure on X. A filter and a filter base on X are examples of structures. Also any ideal of subsets of X is a structure. In this paper several structures are classified and the binary relations and operations on structures are discussed. Furthermore structures on a topological space are also discussed.

Single Valued Neutrosophic Filters

In this paper we give a comprehensive presentation of the notions of filter base, filter and ultrafilter on single valued neutrosophic set and we investigate some of their properties and relationships. More precisely, we discuss properties related to filter completion, the image of neutrosophic filter base by a neutrosophic induced mapping and the infimum and supremum of two neutrosophic filter bases.

On (L,M)-Double Fuzzy Filter Spaces

We give in this paper the definitions of (L,M)-double fuzzy filter base and (L,M)-double fuzzy filter structures where L and M are strictly two-sided commutative quantales, and we also investigate the relations between them. Moreover, we propose second-order image and preimage operators of (L,M)-double fuzzy filter base and study some of its fundamental properties. Finally, we handle the given structures in the categorical aspect. For instance, we show that the category (L,M)-DFIL of (L,M)-double fuzzy filter spaces and filter maps between these spaces is a topological category over the category SET.

Convergent Filter Bases

We are inspired by the work of Henri Cartan [16], Bourbaki [10] (TG. I Filtres) and Claude Wagschal [34]. We define the base of filter, image filter, convergent filter bases, limit filter and the filter base of tails (fr: filtre des sections).