alexa Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 12 2 R -Spaces | OMICS International | Abstract
ISSN: 2168-9679

Journal of Applied & Computational Mathematics
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Research Article

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

Ahmed Saeed Abd-Allah1* and A Al-Khedhairi2

1Department of Statistics and Operations Research, College of Science, King Saud University, Saudi Arabia

2Department of Mathematics, College of Science, El-Mansoura University, El-Mansoura, Egypt

*Corresponding Author:
Ahmed Saeed Abd-Allah
Prince Sattam Bin Abdul-Aziz University
Hotat Bani Tamim, Kingdom of Saudi Arabia
Tel: 00966552057393
E-mail: [email protected]

Received date: January 03 , 2017; Accepted date: April 21, 2017; Published date: April 28, 2017

Citation: Abd-Allah AS, Al-Khedhairi A (2017) Initial and Final Characterized Fuzzy and Finer Characterized Fuzzy -Spaces. J Appl Computat Math 6: 350. doi: 10.4172/2168-9679.1000350

Copyright: © 2017 Abd-Allah AS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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 ϕ 12 -space and 1 3, 2 T ϕ 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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