Ditopological texture spaces and fuzzy topology, II. Topological considerations

Convergence in graded ditopological texture spaces

“…We recall various concepts and properties from [3,4,5] under the following subtitle. Ditopological Texture Spaces: Let S be a set.…”

“…The concept of ditopology is more general than general topology, bitopology and fuzzy topology in Chang's sense. An adequate introduction to the theory of texture spaces and ditopological texture spaces may be obtained from [1,2,3,4,5].…”

$H(i)$ Connected ditopological texture space

“…Different fuzzy topological spaces have been studied by Tripathy and Debnath [19] , Tripathy and Ray [21,20] . A detailed analysis of the relation between texture space and the lattice of fuzzy sets of various kind is found in the works due to [1,4,5,6,3]. The concept of ditopological texture space is introduced by Brown [15].…”

β-open and β-closed sets in ditopological texture spaces

“…Textures and ditopological texture spaces were first introduced by the second author as a point-based setting for the study fuzzy sets, and this line of investigation continues, see for example [5,6,8,9,10], and more recently [23]. On the other hand, textures offer a convenient setting for the investigation of complement-free concepts in general, so much of the recent work has proceeded independently of the fuzzy setting.…”

“…To complete this introduction we recall various concepts from [8,9] that will be needed later on in this paper.…”