International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 423-430
Published online December 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.4.423
© The Korean Institute of Intelligent Systems
Gautam Chandra Ray and Hari Prasad Chetri
Department of Mathematics, Central Institute of Technology, Kokrajhar, Assam, India
Correspondence to :
Gautam Chandra Ray (gautomofcit@gmail.com)
^{*}These authors contributed equally to this work.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Herein, we define fuzzy T_{0}-space, fuzzy T_{1}-space, fuzzy T_{2} (or Hausdorff), as well as fuzzy regular and fuzzy normal spaces in mixed fuzzy topological spaces, and then establish relationships among these spaces. We provide some results for the abovementioned spaces in mixed fuzzy topological spaces.
Keywords: Fuzzy topological spaces, Mixed fuzzy topology, Separation Axioms, T_{0}-space, T_{1}-space, T_{2} (or Hausdorff) spaces
The concept of mixing two topologies to obtain a new topology is not a new concept. In 1958, Alexiewicz and Semadeni [1] introduced and investigated a mixed topology on a set via norm spaces. Thereafter, many mathematicians worldwide investigated mixed topologies and discovered some interesting and applicable results regarding mixed topologies. In this regard, some remarkable results have been reported by Cooper [2], Buck [3], Wiweger [4], and many others. A new era began after the inception of the fuzzy set theory in 1965 by Zadeh [5]. Since the inception of the notion of fuzzy sets and fuzzy logic, fuzziness has been applied to studies in almost all branches of science and technology. The notion of fuzziness has been applied to topology for the first time by Chang [6], and fuzzy topological spaces have been introduced and then investigated extensively. The concept of strong separation and strong countability in fuzzy topological spaces was introduced and investigated by Stadler and de Prada Vicente [7]. In 1995, the concept of mixed fuzzy topological spaces was investigated from different perspectives by Das and Baishya [8]. Recently, in 2012, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology introduced by Das and Baishya [8], by replacing fuzzy points with fuzzy sets. Tripathy and Ray [9] introduced the concept of countability base on a mixed fuzzy topology and proved the existence of three types of countability. However, this concept of mixed fuzzy topology is not a generalization of the concept of a crispy mixed topology (or simply mixed topology) by Wiweger [4], Alexiewicz and Semadeni [1], Cooper [2], and others. Ghanim et. al [10] investigated separation axioms. Recently, W. F. Al-Omeri [11] investigated a mixed b-fuzzy topology using the definition by Das and Baishya [8].
Most recently, Al-Shami and his colleagues [12–21] investigated separation axioms in topological spaces and obtained some important findings. Rashid and Ali [22] introduced separation axioms in a mixed fuzzy topology using the concept of mixed fuzzy topology defined by Das and Baishya [8]. Recently, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology by Das and Baishya [8]. In this study, we redefined and investigated separation axioms in mixed fuzzy topological spaces introduced by Tripathy and Ray [9]. We investigated a few properties of separation axioms that varied slightly, as introduced by Rashid and Ali [22]. We have previously proven some results and constructed a few concrete numerical examples of
Let
The union and intersection of a set {
and
Let
A fuzzy topology on
1̄, 0̄ ∈
For any finite subset
For any arbitrary set Δ of members of
The pair (
The closure of a fuzzy set
A fuzzy set in
A fuzzy point
Two fuzzy sets
A fuzzy set
A fuzzy point
A fuzzy set
It is clear that if
A fuzzy set
Let (
A fts (
A fts (
A fts (
A fts (
fts (
A fts (
Let
Let (
Let
Because every fuzzy point is also a fuzzy singleton set, the theorem is valid with respect to the definition in Theorem 2.
In this section, we analogously define the basic definition of
A mixed fts (
A mixed fts (
A mixed fts (
A mixed fts (
A mixed fts (
Let us consider the following example of mixed fuzzy topology, which comprises fuzz
Let
and
are fuzzy Hausdorff topological spaces. We constructed
Therefore
Therefore,
The complement of {(
However, the complement of {(
Similarly, {(
Therefore,
Next, we prove that
For
Additionally, {(
Therefore,
Let
Clearly,
Next, we consider a mixed fuzzy topological space that is fuzzy
Let
Therefore,
Because every open fuzzy set in
Next, we demonstrate that for every pair
Therefore,
However, {(
Therefore,
Next, we establish the following theorems:
If
Let
Therefore,
Hence, the proof is completed.
If
If
Furthermore,
1) We prove that
Let
2) Next, we prove that
Let
3) Finally, we prove that
Let
⇒
⇒
⇒
If
We assume
Let
Without loss of generality, let an open fuzzy set
Since
Hence,
If
Let us assume that
Let
If
Let
Because
Hence,
Hence,
Let
Let
Therefore,
⇒ 1 −
⇒ 1 −
⇒
⇒open fuzzy sets
Next, for any fuzzy set
⇒
⇒
⇒
The fuzzy regularity of
Therefore,
Therefore,
Furthermore,
⇒ open fuzzy sets
Hence, (
For any non-empty crisp set
Next, we consider a mixed fuzzy topology comprising fuzzy regular and fuzzy normal.
Let
Next, we prove the fuzzy normality and fuzzy regularity of
For the fuzzy regularity, let
If
Next, if
Therefore,
Furthermore,
⇒ {(
Next, we prove that
Let
⇒
⇒
Therefore,
Furthermore,
Therefore,
If
Since
A fuzzy
Let (
Let
Let (
Let
In this study, we defined fuzzy-
No potential conflict of interest relevant to this article was reported.
E-mail: gautomofcit@gmail.com
E-mail: hariprasadchetri1234@gmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 423-430
Published online December 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.4.423
Copyright © The Korean Institute of Intelligent Systems.
Gautam Chandra Ray and Hari Prasad Chetri
Department of Mathematics, Central Institute of Technology, Kokrajhar, Assam, India
Correspondence to:Gautam Chandra Ray (gautomofcit@gmail.com)
^{*}These authors contributed equally to this work.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Herein, we define fuzzy T_{0}-space, fuzzy T_{1}-space, fuzzy T_{2} (or Hausdorff), as well as fuzzy regular and fuzzy normal spaces in mixed fuzzy topological spaces, and then establish relationships among these spaces. We provide some results for the abovementioned spaces in mixed fuzzy topological spaces.
Keywords: Fuzzy topological spaces, Mixed fuzzy topology, Separation Axioms, T_{0}-space, T_{1}-space, T_{2} (or Hausdorff) spaces
The concept of mixing two topologies to obtain a new topology is not a new concept. In 1958, Alexiewicz and Semadeni [1] introduced and investigated a mixed topology on a set via norm spaces. Thereafter, many mathematicians worldwide investigated mixed topologies and discovered some interesting and applicable results regarding mixed topologies. In this regard, some remarkable results have been reported by Cooper [2], Buck [3], Wiweger [4], and many others. A new era began after the inception of the fuzzy set theory in 1965 by Zadeh [5]. Since the inception of the notion of fuzzy sets and fuzzy logic, fuzziness has been applied to studies in almost all branches of science and technology. The notion of fuzziness has been applied to topology for the first time by Chang [6], and fuzzy topological spaces have been introduced and then investigated extensively. The concept of strong separation and strong countability in fuzzy topological spaces was introduced and investigated by Stadler and de Prada Vicente [7]. In 1995, the concept of mixed fuzzy topological spaces was investigated from different perspectives by Das and Baishya [8]. Recently, in 2012, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology introduced by Das and Baishya [8], by replacing fuzzy points with fuzzy sets. Tripathy and Ray [9] introduced the concept of countability base on a mixed fuzzy topology and proved the existence of three types of countability. However, this concept of mixed fuzzy topology is not a generalization of the concept of a crispy mixed topology (or simply mixed topology) by Wiweger [4], Alexiewicz and Semadeni [1], Cooper [2], and others. Ghanim et. al [10] investigated separation axioms. Recently, W. F. Al-Omeri [11] investigated a mixed b-fuzzy topology using the definition by Das and Baishya [8].
Most recently, Al-Shami and his colleagues [12–21] investigated separation axioms in topological spaces and obtained some important findings. Rashid and Ali [22] introduced separation axioms in a mixed fuzzy topology using the concept of mixed fuzzy topology defined by Das and Baishya [8]. Recently, Tripathy and Ray [9] generalized the definition of mixed fuzzy topology by Das and Baishya [8]. In this study, we redefined and investigated separation axioms in mixed fuzzy topological spaces introduced by Tripathy and Ray [9]. We investigated a few properties of separation axioms that varied slightly, as introduced by Rashid and Ali [22]. We have previously proven some results and constructed a few concrete numerical examples of
Let
The union and intersection of a set {
and
Let
A fuzzy topology on
1̄, 0̄ ∈
For any finite subset
For any arbitrary set Δ of members of
The pair (
The closure of a fuzzy set
A fuzzy set in
A fuzzy point
Two fuzzy sets
A fuzzy set
A fuzzy point
A fuzzy set
It is clear that if
A fuzzy set
Let (
A fts (
A fts (
A fts (
A fts (
fts (
A fts (
Let
Let (
Let
Because every fuzzy point is also a fuzzy singleton set, the theorem is valid with respect to the definition in Theorem 2.
In this section, we analogously define the basic definition of
A mixed fts (
A mixed fts (
A mixed fts (
A mixed fts (
A mixed fts (
Let us consider the following example of mixed fuzzy topology, which comprises fuzz
Let
and
are fuzzy Hausdorff topological spaces. We constructed
Therefore
Therefore,
The complement of {(
However, the complement of {(
Similarly, {(
Therefore,
Next, we prove that
For
Additionally, {(
Therefore,
Let
Clearly,
Next, we consider a mixed fuzzy topological space that is fuzzy
Let
Therefore,
Because every open fuzzy set in
Next, we demonstrate that for every pair
Therefore,
However, {(
Therefore,
Next, we establish the following theorems:
If
Let
Therefore,
Hence, the proof is completed.
If
If
Furthermore,
1) We prove that
Let
2) Next, we prove that
Let
3) Finally, we prove that
Let
⇒
⇒
⇒
If
We assume
Let
Without loss of generality, let an open fuzzy set
Since
Hence,
If
Let us assume that
Let
If
Let
Because
Hence,
Hence,
Let
Let
Therefore,
⇒ 1 −
⇒ 1 −
⇒
⇒open fuzzy sets
Next, for any fuzzy set
⇒
⇒
⇒
The fuzzy regularity of
Therefore,
Therefore,
Furthermore,
⇒ open fuzzy sets
Hence, (
For any non-empty crisp set
Next, we consider a mixed fuzzy topology comprising fuzzy regular and fuzzy normal.
Let
Next, we prove the fuzzy normality and fuzzy regularity of
For the fuzzy regularity, let
If
Next, if
Therefore,
Furthermore,
⇒ {(
Next, we prove that
Let
⇒
⇒
Therefore,
Furthermore,
Therefore,
If
Since
A fuzzy
Let (
Let
Let (
Let
In this study, we defined fuzzy-