Neighborhood structures of graded ditopological texture spaces

Abstract

Graded ditopological texture spaces have been presented and discussed in categorical aspects by Alexander Sostak and Lawrence M. Brown in [18]. In this paper, the authors study the generalization of two sorts of neighborhood structure defined in [10] to the graded ditopological texture spaces.

Convergence in graded ditopological texture spaces

“…Graded Dineighborhood Systems: [8] Let (S, S) and (V, V) be two texture spaces. For any mapping K : S → V, we use the notation v K to denote the family {A ∈ S : K(A) Q v } for all v ∈ V , and thus for each v ∈ V , we have…”

Graded diextremities

“…Graded ditopology is more general than ditopology and fuzzy topology inŠostak's sense. Two sorts of neighborhood structure on graded ditopological texture spaces are presented and investigated in [9].…”

“…The difunction ( f, F) is called bicontinuous with respect to (h, H) if it is both continuous and cocontinuous with respect to (h, H). The graded dineighborhood systems of the graded ditopological texture spaces were defined in [9]. From now on, we will use dinhd, shortly instead of dineighborhood.…”