Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 12 2 R -Spaces

Journal of Applied and Computational Mathematics Pub Date : 2017-04-28 DOI:10.4172/2168-9679.1000350

A. Abd-Allah, A. Al-khedhairi

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Abstract

Basic notions related to the characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces are introduced and studied. The metrizable characterized fuzzy spaces are classified by the characterized fuzzy 1 2 2 R and the characterized fuzzy T4-spaces in our sense. The induced characterized fuzzy space is characterized by the characterized fuzzy 1 3 2 T and characterized fuzzy 1 3 2 T -space if and only if the related ordinary topological space is 1 2, 2 R I 12 -space and 1 3, 2 T I 12 -space, respectively. Moreover, the α-level and the initial characterized spaces are characterized 1 2 2 R and characterized 1 3 2 T -spaces if the related characterized fuzzy space is characterized fuzzy 12 2 R and characterized fuzzy 1 3 2 T , respectively. The categories of all characterized fuzzy 1 2 2 R and of all characterized fuzzy 1 3 2 T -spaces will be denoted by CFR-Space and CRF-Tych and they are concrete categories. These categories are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych. That is, all the initial and final characterized fuzzy 1 2 2 R spaces and all the initial and final characterized fuzzy 1 3 2 T -spaces exist in CFR-Space and in CRF-Tych. The initial and final characterized fuzzy spaces of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. As special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are also characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. Finally, three finer characterized fuzzy 1 2 2 R -spaces and three finer characterized fuzzy 1 3 2 T -spaces are introduced and studied.

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初始和最终表征模糊1 3 2 T和精细表征模糊12 2 R -空间

介绍并研究了特征模糊1 2 2 R和特征模糊1 3 2 T空间的基本概念。可度量的特征模糊空间由我们意义上的特征模糊1 2 2 R和特征模糊t4空间进行分类。当且仅当相关普通拓扑空间分别为1,2,2ri - 12空间和1,3,2ti - 12空间时，导出的表征模糊空间分别为表征模糊1,3,2ri - 12空间和表征模糊1,32t - T空间。如果相关特征模糊空间的特征为模糊122r和模糊132t，则α-层和初始特征空间分别为1 22r和1 322t。将所有表征模糊1 2 2 R和所有表征模糊1 3 2 T -空间的范畴用CFR-Space和CRF-Tych表示，它们是具体的范畴。这些范畴是所有特征模糊空间的范畴cf -空间的满子范畴，它们在所有子集的范畴集合上是拓扑的，因此所有的初升和终升在cfr -空间和CRF-Tych中唯一存在。即在CFR-Space和CRF-Tych中存在所有的初始和最终表征模糊1 2 2 R空间以及所有的初始和最终表征模糊1 3 2 T空间。表征模糊1 2 2 R -空间和表征模糊1 3 2 T -空间的初始和最终表征模糊空间分别表征为模糊1 2 2 R和表征模糊1 3 2 T -空间。作为特殊情况，表征模糊1 2 2 R空间和表征模糊1 3 2 T空间的表征模糊子空间、表征模糊积空间、表征模糊商空间和表征模糊和空间也分别表征模糊1 2 2 R和表征模糊1 3 2 T空间。最后介绍并研究了3个更精细的特征模糊1 2 2 R空间和3个更精细的特征模糊1 3 2 T空间。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Journal of Applied and Computational Mathematics

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