International Journal of Electrical and Computer Engineering (IJECE)
Vol. 8, No. 6, December 2018, pp. 4619~4625
ISSN: 2088-8708, DOI: 10.11591/ijece.v8i6.pp4619-4625 4619
Journal homepage: http://iaescore.com/journals/index.php/IJECE
Fuzzy Homogeneous Bitopological Spaces
Samer Al Ghour, Almothana Azaizeh
1
Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan
2
College of Applied Studies and Community Service, Imam Abdurrahman Bin Faisal University, Saudi Arabia
Article Info ABSTRACT
Article history:
Received Feb 15, 2018
Revised May 28, 2018
Accepted Jun 11, 2018
We continue the study of the concepts of minimality and homogeneity in the
fuzzy context. Concretely, we introduce two new notions of minimality in
fuzzy bitopological spaces which are called minimal fuzzy open set and
pairwise minimal fuzzy open set. Several relationships between such notions
and a known one are given. Also, we provide results about the transformation
of minimal, and pairwise minimal fuzzy open sets of a fuzzy bitopological
space, via fuzzy continuous and fuzzy open mappings, and pairwise
continuous and pairwise open mappings, respectively. Moreover, we present
two new notions of homogeneity in the fuzzy framework. We introduce the
notions of homogeneous and pairwise homogeneous fuzzy bitopological
spaces. Several relationships between such notions and a known one are
given. And, some connections between minimality and homogeneity are
given. Finally, we show that cut bitopological spaces of a homogeneous
(resp. pairwise homogeneous) fuzzy bitopological space are homogeneous
(resp. pairwise homogeneous) but not conversely.
Keyword:
Cut topologies
Fuzzy bitopology
Fuzzy homeomorphism
Homogeneous topology
Minimal fuzzy open set
Copyright © 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Samer Al Ghour,
Department of Mathematics and Statistics,
Jordan University of Science and Technology, Jordan.
Email: algore@just.edu.jo
1. INTRODUCTION
Throughout this paper, will denote the interval [0,1]. Let be a nonempty set. A member of
is
called a fuzzy subset of [1]. Throughout this paper, for , ∈
we write ≤ iff () ≤ () for all
∈ . By = we mean that ≤ and ≤, i.e., () = () for all ∈ . Also we write <
iff ≤ and ≠. If {
: ∈ } is a collection of fuzzy sets in , then (⋁
) () = {
(): ∈
}, ∈ ; (⋀
) () = {
(): ∈ }, ∈ . ∈ [0,1] then
denotes the fuzzy set given by
() = for all ∈ . If ⊆ then
denotes the characteristic function of . A fuzzy set
defined by
() = {
, =
0, ≠
where 0<≤1 is called a fuzzy point in ,
∈ is called the support of and (
)= the value
(level) of [2]. In this paper, a fuzzy point in is said to belong to a fuzzy set in [3] (notation: ∈
) iff (
) ≤ (
).
Let : → be an ordinary mapping. We define
→
∶
→
and
←
∶
→