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The aim of this paper is to introduce two different types of connectedness notions for graded ditopological texture spaces: the connectedness function which gives the grade of connectedness of a set and the connectedness spectrum by means of spectrum idea. Also, the properties of these connectedness notions and their relationships with the connectedness notion in ditopological case are investigated. Further, the relation between these two different types of connectedness notions is studied.

In this paper, we introduce the fundamental aspect of H(i) connected ditopological texture space, before proving we develop some basic key of bicontinuity and connectedness in term of ditopological texture space which will helpful in H(i) connected ditopological texture space. We establish some correspondence related to known structure such as bitopological space, fuzzy lattice and topological space.

This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits.

− The main purpose of this paper, is to introduce the notion of ditopological texturespaces with ideal. We study the notions of ?-connected ditopological texture spaces with an idealand ?-connected sets in ditopological texture spaces with ideal. Some new types of connectedness in?-ditopological texture spaces namely, locally ?-connectedness, totally ?-disconnectedness, hyperconnectedness and ?-hyperconnectedness have investigated