Graded ditopological texture spaces have been presented and discussed in categorical aspects by Lawrence M. Brown and AlexanderŠostak in [6]. In this paper, the authors generalize the structure of difilters in ditopological texture spaces defined in [11] to the graded ditopological texture spaces and compare the properties of difilters and graded difilters.2010 MSC: 54A05; 54A20; 06D10.
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.
In this paper, the concept of graded diextremity is defined on textures as a generalization of diextremities on textures and some properties of graded diextremity are obtained. It is shown that each graded diuniformity generates a graded diextremity and each graded diextremity genarates a graded ditopology. Moreover, the relations between graded diextremities (resp. graded diuniformities, graded ditopologies) and diextremities (resp. diuniformities, ditopologies) are investigated in basic categorical aspects.
In this paper, initial and product graded ditopologies are formulated and accordingly it is shown that dfGDitop is a topological structure over dfTex dfTex. By means of spectrum idea, (di)compactness in graded ditological texture spaces is defined as a generalization of (di)compactness in ditopological case and its relation with the ditopological case is investigated. Moreover, the relations between graded difilters and dicompactness of graded ditological texture spaces are studied.
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