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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

Separation Axioms in Mixed Fuzzy Topological Spaces

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.

Received: January 3, 2021; Revised: September 12, 2021; Accepted: September 15, 2021

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 T0-space, fuzzy T1-space, fuzzy T2 (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, T0-space, T1-space, T2 (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 [1221] 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 T0, T1, T2(Hausdorff), and regular and normal mixed fuzzy topological spaces such that the spaces can be visualized more effectively. Herein, we provide the basic definitions in Section 2, although most of the concepts, definitions, and notations used herein are standard ones.

Definition 1 [23]

Let X be a non-empty set, and I be the unit interval [0, 1]. A fuzzy set A in X is characterized by a function μA : XI, where μA is known as the membership function of A. μA(x) representing the membership grade of x in A. The empty fuzzy set is defined as μφ(x) = 0 for all xX. Additionally, X can be regarded as a fuzzy set in itself, defined μX(t) = 1 for all tX. Furthermore, an ordinary subset A of X can be regarded as a fuzzy set in X if its membership function is regarded as the usual characteristic function of A, i.e., μA(x) = 1, for all xX and μA(x) = 0 for all xXA. The two fuzzy sets A and B are equal if μA = μB. A fuzzy set A is contained in a fuzzy set B, written as AB, if μAμB. The complement of a fuzzy set A in X is a fuzzy set A in X, defined as μAc = 1− μA. We write Ac = coA to avoid confusion.

The union and intersection of a set {Ai : iI} of fuzzy sets in X to be written as i=1Ai and i=1Ai, separately, are defined as follows:

μiIAi(x)=sup{μAi(x):iI}for all xX,

and

μiIAi(x)=inf{μAi(x):iI}for all xX.

Definition 2 [6]

Let I = [0, 1], X be a non-empty set, and IX be a set of all mappings from X to I, i.e., the class of all fuzzy sets in X.

A fuzzy topology on X is a family τ of member IX, such that

  • 1̄, 0̄ ∈ τ.

  • For any finite subset B={Bi}i=1n of members of τ, i=1n{Bi}τ.

  • For any arbitrary set Δ of members of τ, ∪BΔ {B} ∈ τ.

The pair (X, τ ) is known as a fuzzy topological space (fts), and the members of τ are known as τ-open fuzzy sets.

Definition 3 [24]

The closure of a fuzzy set A in an fts (X, τ ) is defined as the intersection of all closed supersets of A, i.e.,

A¯={F:AF,Fis a closed fuzzy set}.

Definition 4 [23]

A fuzzy set in X is known as a fuzzy point if and only if it assumes the value 0 for all yX except one, e.g., xX. If its value at x is α (0 <≤ 1), then this point is denoted by x, and we refer to point x as its support.

Definition 5 [23]

A fuzzy point xλ is contained in a fuzzy set A or belongs to A, denoted by xαA, if and only if αA(x).

Definition 6 [23]

Two fuzzy sets A and B in X are intersecting if a point xX exists such that (AB)(x) ≠ 0. In this case, A and B intersect at x.

Definition 7 [23]

A fuzzy set A in a fts (X, τ ) is known as the neighborhood of a fuzzy point xλX if and only if Bτ exists such that xλBA; a neighborhood A is known as an open neighborhood if and only if A is fuzzy open. A family comprising all the neighborhoods of xλ is known as a system of neighborhoods of xλ.

Definition 8 [23]

A fuzzy point xα is quasi-coincident with a fuzzy set A, denoted by xαqA, if and only if α + A(x) > 1 or α > (A(x))c.

Definition 9 [23]

A fuzzy set A is quasi-coincident with B and is denoted by AqB if and only if xX exists such that A(x) + B(x) > 1.

It is clear that if A and B are quasi-coincident at x, then both A(x) and B(x) are non-zero at x; hence, A and B intersect at x.

Definition 10 [23]

A fuzzy set A in an fts (X, τ ) is known as a quasi-neighborhood (Q-neighborhood) of xλ if and only if ∃ A1τ, such that A1A and xλqA1. A family Uxλ comprising all the Q-neighborhoods of x is known as the Q-neighborhood system of x. The intersection of two Q-neighborhoods of xλ is a quasi-neighborhood.

Theorem 1 [8]

Let (X, τ1) and (X, τ2) be two fuzzy topological spaces. Consider fuzzy sets τ1(τ2) = {AIX: For any xqA, a τ2-Q-neighborhood A of x exists such that τ1-closure Ā|αA}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Definition 11 [10]

A fts (X, τ ) is known as a fuzzy T0-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ such that xαUco(yβ) or yβVco(xα) in τ.

Definition 12 [10]

A fts (X, τ ) is known as a fuzzy T1-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ such that xαU and yβU, and xαV and yβV.

Definition 13 [10]

A fts (X, τ ) is known as the fuzzy Hausdorff or fuzzy T2-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist such that xαU and yβV, and UV = 0̄.

Definition 14 [10]

A fts (X, τ ) is known as fuzzy regular if and only if for any fuzzy point x and a closed fuzzy set B with xBc, open fuzzy sets U and V exist in τ such that xαU and BV, and U ⊆ 1 − V.

Proposition 1 [25]

fts (X, τ ) is known as a fuzzy regular space if and only if for any fuzzy point xα in IX and Aτ withα < A(x), Bτ exists such that α < B(x) and A.

Definition 15 [22]

A fts (X, τ ) is known as a fuzzy normal space if and only if for each closed set B and an open fuzzy set U in (X, τ ) where BU, Vτ exists such that BVVU.

Lemma 1 [8]

Let τ1 and τ2 be two fuzzy topological spaces on set X. If every τ1-Q-neighbourhood of x is a τ2-Q-neighbourhood of x for all fuzzy points x, then τ1 is coarser than τ2.

Theorem 2 [9]

Let (X, τ1) and (X, τ2) be two fuzzy topological spaces. Consider fuzzy sets τ1(τ2) = {AIX: For any fuzzy set B in X with AqB, a τ2-open set A1 exists such that A1qB and τ1-closure Ā1A}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Theorem 3 [8]

Let τ1 and τ2 be two fuzzy topological spaces on set X. Subsequently, the mixed fuzzy topology τ1(τ2) is coarser than τ2. This is represented as τ1(τ2) ⊆ τ2.

Proof

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 T0, T1, T2 (or Hausdorff), as well as regular and normal mixed fuzzy topological spaces (X, τ1(τ2)), and then provide some results in mixed fuzzy topological spaces. The definitions used herein are as follows:

Definition 16

A mixed fts (X, τ1(τ2)) is known as a fuzzy T0-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ1(τ2) such that xαUco(yβ) or yβVco(xα) in τ1(τ2).

Definition 17

A mixed fts (X, τ1(τ2)) is known as a mixed fuzzy T1-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V in τ1(τ2) exist such that xαU and yβU as well as xαV and yβV.

Definition 18

A mixed fts (X, τ1(τ2)) is known as a fuzzy Hausdorff or fuzzy T2-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ1(τ2) such that xαU and yβV, and UV = 0̄.

Definition 19

A mixed fts (X, τ1(τ2)) is known as a fuzzy regular space if and only if for any fuzzy point xα in IX and a closed set B in τ1(τ2) with xBc, open fuzzy sets U and V exist in τ1(τ2) such that xαU and BV, and U ⊆ 1 − V.

Definition 20

A mixed fts (X, τ1(τ2)) is known as a fuzzy normal space if and only if for each closed fuzzy set B and an open fuzzy set U in τ1(τ2) where BU, Vτ1(τ2) exists such that BVU.

Example 1

Let us consider the following example of mixed fuzzy topology, which comprises fuzz T0, fuzzy T1 and fuzzy T2.

Let X = {x, y} and xy; therefore

τ1={0¯,1¯,{(x,α),(y,0)},{(x,0),(y,α)},{(x,α),(y,β)}|α,β[0.6,1]},

and

τ2={0¯,1¯,{(x,0.3),(y,0)},{(x,0),(y,0.3)},{(x,0.5),(y,0)},{(x,0),(y,0.5)},{(x,0.3),(y,03)},{(x,0.5),(y,0.5)},{(x,1),(y,0)},{(x,0),(y,1)}}

are fuzzy Hausdorff topological spaces. We constructed τ1(τ2) from these topologies. In addition, τ1(τ2) ⊆ τ2. Therefore we verified the belongingness of {(x, 0.3), (y, 0.3)}, {(x, 0.5), (y, 0.5)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x, 0.5), (y, 0)}, {(x, 0), (y, 0.5)}, {(x, 1), (y, 0)}, and {(x, 0), (y, 1)} in τ1(τ2). As an example, = {(x, 0.3), (y, 0.3)} ∈ τ2 for A1 = {(x, δ): δ > 0.7}; AqA1 and the τ1 closure of (A = {(x, 0.3), (y, 0.3)}) = AA. Because {(x, 0.7), (y, 0.7)} is an open set in τ1, { (x, 0.3), (y, 0.3)} is a closed set in τ1. (Similarly, for A2 = {(y, δ): δ > 0.7} or A3 = {(x, γ), (y, δ) | γ, δ ∈ (0.7, 1]}; AqA2 and AqA3 are also a τ1 closure of (A = {(x, 0.3), (y, 0.3)}) = AA.)

Therefore Aτ1(τ2). However, for B = {(x, 0.5), (y, 0.5)} ∈ τ2 and A2 = {(x, δ): δ > 0.5} BqA2 and the τ1 closure of (B = {(x, 0.5), (y, 0.5)}) = 1̄ ⊄ B.

Therefore, Bτ1(τ2). Similarly, as in the case of A = {(x, 0.3), (y, 0.3)} ∈ τ2, {(x, 1), (y, 0)} and {(x, 0), (y, 1)} will be members of τ1(τ2). Furthermore, τ1 is the closure of {(x, 1), (y, 0)} = {(x, 1), (y, 0)}, as {(x, 0), (y, 1)} is an open set in τ1; therefore, the complement {(x, 1), (y, 0)} is closed. In addition, the τ1 closure of {(x, 0), (y, 1)} = {(x, 0), (y, 1)} as {(x, 1), (y, 0)} is an open set in τ1.

The complement of {(x, 0.3), (y, 0)} in τ1 is {(x, 0.7), (y, 1)}, which is an open set such that {(x, 0.3), (y, 0)} is a closed set in τ1. Therefore, as in the case of A = {(x, 0.3), (y, 0)} ∈ τ1(τ2), we have {(x, 0), (y, 0.3)} ∈ τ1(τ2).

However, the complement of {(x, 0.5), (y, 0)} in τ1 is {(x, 0), (y, 0.5)} and not fuzzy open in τ1; therefore, {(x, 0.5), (y, 0)} is not fuzzy closed, and the τ1-closure of {(x, 0.5), (y, 0)} ⊄ {(x, 0.5), (y, 0)}.

Similarly, {(x, 0.5), (y, 0)}∉ τ1(τ2) and {(x, 0), (y, 0.5)} ∉ τ1(τ2).

Therefore, τ1(τ2) = {0̄, 1̄, {(x, 0.3), (y, 0.3)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x 1), (y, 0)}, {(x, 0), (y, 1)}}.

Next, we prove that τ1(τ2) is fuzzy T0, fuzzy-T1, and fuzzy-T2.

For xα and yβ in IX, xα ∈ {(x, 1), (y, 0)} ⊆ co(yβ) and yβ ∈ {(x, 0), (y, 1)} ⊆ co(xα).

Additionally, {(x, 1), (y, 0)} ∩ {(x, 0), (y, 1)} = 0̄.

Therefore, τ1(τ2) qualifies as fuzzy-T0, fuzzy-T1, and fuzzy-T2.

Example 2

Let X = {x, y}; subsequently, τ1 = {1̄, 0̄, {(x, 0), (y, 1)}} and τ2 = {1̄, 0̄, {(x, 1), (y, 0)}, {(x, 0.7), (y, 0.7)}, {(x, 0.7), (y, 0)}, {(x, 1), (y, 0.7)}} are fuzzy topologies. Therefore, τ1(τ2) = {1̄, 0̄, {(x, 1), (y, 0)}}, since {(x, 1), (y, 0)} is a closed fuzzy set in τ1. For any fuzzy set B in X with {(x, 1), (y, 0)}qB, we have a τ2-open fuzzy set {(x, 1), (y, 0)} such that {(x, 1), (y, 0)}qB and τ1-closure ({(x,1),(y,0)}¯){(x,1),(y,0)}.

Clearly, τ1(τ2) is mixed fuzzy T0 but not mixed fuzzy T1, as xα ∈ {(x, 1), (y, 0)} ⊆ co(yβ) and ∄ Aτ1(τ2) such that yαAco(xβ).

Next, we consider a mixed fuzzy topological space that is fuzzy T1 but not fuzzy T2.

Example 3

Let X = {x, y, z}; subsequently, τ1 = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}, {(x, 0.7), (y, 0)(z, 0)}, {(x, 0.7), (y, 1)(z, 0)}, {(x, 0.7), (y, 0)(z, 1)}, {(x, 0.7), (y, 1)(z, 1)}} and τ2 = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1)(y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}} are fuzzy topologies.

Therefore, τ1(τ2) = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}}.

Because every open fuzzy set in τ2 is a closed fuzzy set in τ1. Therefore, every open fuzzy set in τ2 is an open fuzzy set in τ1(τ2), as shown in Example 2.

Next, we demonstrate that for every pair xα, yβ, yα, zβ, or zα, xβ, open fuzzy sets U and V exist in τ1(τ2) such that xαUco(yβ) and yβVco(xα). We will verify this only for the pair xα, yβ; similarly, other cases can be completed. We have open fuzzy sets {(x, 1), (y, 0)(z, 1)} and {(x, 0), (y, 1)(z, 1)} in τ1(τ2) such that xα ∈ {(x, 1), (y, 0)(z, 1)} ⊆ co(yβ) and yβ ∈ {(x, 0), (y, 1)(z, 1)} ⊆ co(xα).

Therefore, τ1(τ2) is fuzzy T1.

However, {(x, 1), (y, 0)(z, 1)} ⊄ co({(x, 0), (y, 1)(z, 1)}).

Therefore, τ1(τ2) is not a mixed fuzzy T2 space.

Next, we establish the following theorems:

Theorem 4

If τ1τ2 and τ1 is fuzzy regular, then τ1τ1(τ2).

Proof

Let A be a τ1-Q-neighborhood of xα, then ∃ A1τ1 such that

xαqA1Aα+A1(x)>1(0<α1)A1(x)>1-α=β(say)A1(x)>β,where 0<β1Bτ1exists such that β<B(x)and τ1-closure of         BA1         (Since τ1is fuzzy regular).

Therefore,

1-α<B(x)1<B(x)+αand τ1-closure B¯A1xαqB,and Bτ2and τ1-closure B¯A1AAτ1(τ2)(By definition of τ1(τ2))Ais a τ1(τ2)quasi-neighborhood of xατ1τ1(τ2).

Hence, the proof is completed.

Theorem 5

If τ1τ2 and τ1 are fuzzy regular, then τ1(τ2) is a fuzzy-T0, fuzzy-T1, and fuzzy-T2 (or Hausdorff) space.

Proof

If τ1 is fuzzy regular, then τ1 is a fuzzy-T0, fuzzy-T1, and fuzzy-T2 (or Hausdorff) space.

Furthermore, τ1τ1(τ2) (by Theorem 4).

1) We prove that τ1(τ2) is a fuzzy T0-space.

Let xα, yβ be any two fuzzy points such that xy. Because τ1 is a fuzzy T0-space, without loss of generality, let an open fuzzy set Uτ1 such that

xαUco(yβ)xαUτ1(τ2)and yβU[since τ1τ1(τ2)]open fuzzy set Uτ1(τ2)such that xαUco(yβ)τ1(τ2)is fuzzy T0space.

2) Next, we prove that τ1(τ2) is a fuzzy T1-space.

Let xα and yβ be any two fuzzy points such that xy. Because τ1 is a fuzzy T1-space, open fuzzy sets V1 and V2 in τ1 exist such that xαV1 and yβV1, and xαV2 and yβV2. Since τ1τ1(τ2), therefore V1 and V2 are in τ1(τ2) such that xαV1 and yβV1, and xαV2 and yβV2. Therefore, τ1(τ2) is a fuzzy T1-pace.

3) Finally, we prove that τ1(τ2) is a fuzzy Hausdorff space.

Let xα and yβ be any two fuzzy points such that xy.

V1 and V2 exist in τ1 such that xαV1 and yβV2 and V1V2 = 0̄ (since τ1 is fuzzy Hausdorff).

V1 and V2 are in τ1(τ2) such that xαV1 and yβV2, and V1V2 = 0̄. [since τ1τ1(τ2)]

τ1(τ2) is a fuzzy Hausdorff space.

Theorem 6

If τ1(τ2) is a fuzzy T0-space, then τ2 is also a fuzzy T0-space.

Proof

We assume τ1(τ2) to be a fuzzy T0-space.

Let xα and yβ be any two fuzzy points such that xy.

Without loss of generality, let an open fuzzy set Uτ1(τ2) exist such that xαUco(yβ).

Since τ1(τ2) ⊆ τ2, Uτ2 such that xαUco(yβ).

Hence, τ2 is a fuzzy T0-space.

Theorem 7

If τ1(τ2) is a fuzzy T1-space, then τ2 is also a fuzzy T1-space.

Proof

Let us assume that τ1(τ2) is a fuzzy T1-space.

Let xα and yβ be any two fuzzy points such that xy. Because τ1(τ2) is a fuzzy T1-space, open fuzzy sets V1 and V2 exist in τ1(τ2) such that xαV1 and yβV1, xαV2, and yβV2. Since τ1(τ2) ⊆ τ2, V1 and V2 exist in τ2 such that xαV1 and yβV1, and xαV2 and yβV2. Hence, τ2 is also a fuzzy T1-space.

Theorem 8

If τ1(τ2) is a fuzzy T2-space (or Hausdorff), then τ2 is also a fuzzy T2-space.

Proof

Let xα and yβ be any two fuzzy points such that xy.

Because τ1(τ2) is fuzzy Hausdorff, V1 and V2 exist in τ1(τ2) such that xαV1 and yβV2, and V1V2 = 0̄.

Hence, V1 and V2 exist in τ2 such thauto-at let xαV1 and yβV2, and V1V2 = 0̄ (since τ1(τ2) ⊆ τ2).

Hence, τ2 is also a fuzzy T2-space.

Theorem 9

Let τ1τ2 and τ2 be fuzzy regular; therefore, τ1(τ2) is also fuzzy regular.

Proof

Let xα be a fuzzy point and C be a closed fuzzy set in τ1(τ2) such that xαCc.

Therefore, C is a closed fuzzy set in τ1(τ2).

⇒ 1 − C is an open fuzzy set in τ1(τ2).

⇒ 1 − C is an open fuzzy set in τ2 (since τ1(τ2) ⊆ τ2).

C is closed in τ2.

⇒open fuzzy sets A1 and A2 exist in τ2 such that xαA1 and CA2 and A1 ⊆ 1 − A2.

Next, for any fuzzy set B such that BqA1

B(z) + A1(z) > 1, for some zX

A1(z) > 1 − B(z)

A1(z) > r, where r = 1− B(z).

The fuzzy regularity of τ2 guarantees that ∃ A2τ2 such that A2(z) > r and A2τ2cl(AtA1, since τ1τ2.

Therefore,

τ1-cl(A2)={B|A2B,where Bis a closed fuzzy set in τ1}{B|A2B,Bis a closed fuzzy set in τ2}=τ2-cl(A2).[Every closed fuzzy set in τ2is a closed fuzzy set in τ1].

Therefore, A2qB and A2τ1cl(A2) ⊆ A1.

Furthermore, A1τ1(τ2), and similarly, we can show that A2τ1(τ2).

⇒ open fuzzy sets A1 and A2 exist in τ1(τ2) such that xαA1 and CA2, and A1 ⊆ 1 − A2.

Hence, (X, τ1(τ2)) is a fuzzy regular space.

Corollary 1

For any non-empty crisp set X, if τ2τ1(τ2), then τ2 = τ1(τ2) because τ1(τ2) ⊆ τ2.

Next, we consider a mixed fuzzy topology comprising fuzzy regular and fuzzy normal.

Example 4

Let X = {x, y} and xy. Let τ1 be the fuzzy topology generated by the base and τ2 be the fuzzy topology generated by the base , where α is a rational number} ∪ {yβ | β ∈ (0, 1] and β is a rational number} ∪ {∅︀}. Therefore, τ1 and τ2 are fuzzy regular and fuzzy normal spaces, respectively. Next, we show that the mixed fuzzy topology τ1(τ2) is the same as the fuzzy topology τ2. We know that τ1(τ2) ⊆ τ2. Moreover, for any element in the base, e.g., B = {xα | α ∈ (0, 1], where α is a rational number}, the fuzzy set C = {(x, 1 − α), (y, 1)} is an open fuzzy set in τ1 and a complement of B in τ1. Therefore B is a closed fuzzy set in τ1, and τ1-closure of B = B. Therefore, for any fuzzy set A with AqBτ2 open fuzzy set B such that AqB and τ1-closure of B = B. Therefore, Bτ1(τ2). Similarly, any base element {yβ | β ∈ (0, 1], where β is a rational number} is also in τ1(τ2); this implies τ2τ1(τ2). Therefore, τ1(τ2) = τ2.

Next, we prove the fuzzy normality and fuzzy regularity of τ1(τ2).

For the fuzzy regularity, let x be any fuzzy point and B be a closed fuzzy set in τ1(τ2) such that

xBcBc(x)λ1-B(x)λB(x)1-λB(x)=1-λor B(x)<1-λ.

If B(x) = 1 − λ, then 1 − λ is a rational number.

B ⊆ {(x, 1 − λ), (y, 1)} = V is an open fuzzy set in τ1(τ2) and xU = {(x, λ), (y, 0)} ∈ τ1(τ2) such that U = {(x, λ), (y, 0)} ⊆ 1 − V = {(x, λ), (y, 0)}. Therefore, in this case, τ1(τ2) is fuzzy regular.

Next, if B(x) < 1−, then ∃ is a rational number δ such that B(x) < δ < 1 − λ.

Therefore, B ⊆ {(x, δ), (y, 1)} = V is an open fuzzy set in τ1(τ2).

Furthermore,

δ<1-λ1-δ>1-(1-λ)1-δ>λis a rational number rsuch that 1-δ>r>λ

⇒ {(x, r), (y, 0)} = U is an open set in τ1(τ2) such that xU and Vc = {(x, 1 − δ), (y, 0)}, resulting in UVc; therefore, τ1(τ2) is fuzzy regular in both cases.

Next, we prove that τ1(τ2) is fuzzy normal.

Let B be any closed fuzzy set and U be an open fuzzy set such that BU.

B(x) ≤ U(x), ∀ xX and U is an open fuzzy set in τ1(τ2). B(x) = 1 − r for some rational number

r(0,1]B(x)B(x)+U(x)2U(x) and B(x)+U(x)2 is a rational number in (0, 1].

Therefore, V(x)=B(x)+U(x)2 and V(y)=B(y)+U(y)2 defines a fuzzy open set in τ1(τ2).

Furthermore, B(x) ≤ V (x) ≤ U(x) ∀ xX. Similarly, 1−V yields rational values for both x and y; therefore, 1−V is open in τ1(τ2) ⇒ V is fuzzy closed as well as ⇒ V (x) = (x) ∀ xX.

B(x)V(x)V¯(x)U(x)xXB(x)V(x)V¯(x)U(x).

Therefore, τ1(τ2) is a normal mixed fuzzy topological space.

Theorem 10

If τ1τ2 and τ2 are fuzzy normal, then τ1(τ2) is also fuzzy normal.

Proof

Since τ2 is fuzzy normal, it is fuzzy regular. Therefore, from Corollary 1, we obtain τ1(τ2) = τ2. Hence, τ1(τ2) is also normal.

Theorem 11

A fuzzy T2 mixed fts (X, τ1(τ2)) is a mixed fuzzy T1-space, and a mixed fuzzy T1 space is a mixed fuzzy T0-space.

Proof

Let (X, τ1(τ2)) be a T2 mixed fuzzy topological space.

Let xα and yβ be two fuzzy points such that xy. Therefore, open fuzzy sets O1 and O2 exist such that xαO1co(yβ), yβO2co(xα), and O1co(O2). Therefore, xαO1co(yβ) and yβO2co(xα); as such, (X, τ1(τ2)) is a T1 mixed fuzzy topological space.

Let (X, τ1(τ2)) be a T1 mixed fuzzy topological space.

Let xα and yβ be two fuzzy points such that xy. Therefore, open fuzzy sets O1 and O2 exist such that xαO1co(yβ) and yβO2co(xα). Consequently, xαO1co(yβ) or yβO2co(xα). Hence, (X, τ1(τ2)) is a mixed fuzzy T0 topological space.

In this study, we defined fuzzy-T0, fuzzy-T1, fuzzy-T2 (or Hausdorff), fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. We established the relationships among fuzzy-T0, fuzzy-T1, fuzzy-T2, fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. Only a few theorems have been established to investigate separation axioms in mixed fuzzy topological spaces. More properties and theorems pertaining to separation axioms will be established in the near future using these established results in the context of mixed fuzzy topological spaces.

No potential conflict of interest relevant to this article was reported.

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Gautam Chandra Ray received a Ph.D. degree in Mathematics from Gauhati University, Assam, India. He is an assistant professor in the Department of Mathematics in CIT Kokrajhar, Assam, India. His research interests include neutrosophic sets and logic, fuzzy sets and systems, topology, and graph theory.

E-mail: gautomofcit@gmail.com


Hari Prasad Chetri received his M.Sc. degree in mathematics from Gauhati University, India. He is a research scholar in the Department of Mathematics of the Central Institute of Technology, Kokrajhar, India. His research interests include general topology, fuzzy topology, and algebra.

E-mail: hariprasadchetri1234@gmail.com


Article

Original Article

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.

Separation Axioms in Mixed Fuzzy Topological Spaces

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.

Received: January 3, 2021; Revised: September 12, 2021; Accepted: September 15, 2021

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.

Abstract

Herein, we define fuzzy T0-space, fuzzy T1-space, fuzzy T2 (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, T0-space, T1-space, T2 (or Hausdorff) spaces

1. Introduction

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 [1221] 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 T0, T1, T2(Hausdorff), and regular and normal mixed fuzzy topological spaces such that the spaces can be visualized more effectively. Herein, we provide the basic definitions in Section 2, although most of the concepts, definitions, and notations used herein are standard ones.

2. Preliminaries

Definition 1 [23]

Let X be a non-empty set, and I be the unit interval [0, 1]. A fuzzy set A in X is characterized by a function μA : XI, where μA is known as the membership function of A. μA(x) representing the membership grade of x in A. The empty fuzzy set is defined as μφ(x) = 0 for all xX. Additionally, X can be regarded as a fuzzy set in itself, defined μX(t) = 1 for all tX. Furthermore, an ordinary subset A of X can be regarded as a fuzzy set in X if its membership function is regarded as the usual characteristic function of A, i.e., μA(x) = 1, for all xX and μA(x) = 0 for all xXA. The two fuzzy sets A and B are equal if μA = μB. A fuzzy set A is contained in a fuzzy set B, written as AB, if μAμB. The complement of a fuzzy set A in X is a fuzzy set A in X, defined as μAc = 1− μA. We write Ac = coA to avoid confusion.

The union and intersection of a set {Ai : iI} of fuzzy sets in X to be written as i=1Ai and i=1Ai, separately, are defined as follows:

μiIAi(x)=sup{μAi(x):iI}for all xX,

and

μiIAi(x)=inf{μAi(x):iI}for all xX.

Definition 2 [6]

Let I = [0, 1], X be a non-empty set, and IX be a set of all mappings from X to I, i.e., the class of all fuzzy sets in X.

A fuzzy topology on X is a family τ of member IX, such that

  • 1̄, 0̄ ∈ τ.

  • For any finite subset B={Bi}i=1n of members of τ, i=1n{Bi}τ.

  • For any arbitrary set Δ of members of τ, ∪BΔ {B} ∈ τ.

The pair (X, τ ) is known as a fuzzy topological space (fts), and the members of τ are known as τ-open fuzzy sets.

Definition 3 [24]

The closure of a fuzzy set A in an fts (X, τ ) is defined as the intersection of all closed supersets of A, i.e.,

A¯={F:AF,Fis a closed fuzzy set}.

Definition 4 [23]

A fuzzy set in X is known as a fuzzy point if and only if it assumes the value 0 for all yX except one, e.g., xX. If its value at x is α (0 <≤ 1), then this point is denoted by x, and we refer to point x as its support.

Definition 5 [23]

A fuzzy point xλ is contained in a fuzzy set A or belongs to A, denoted by xαA, if and only if αA(x).

Definition 6 [23]

Two fuzzy sets A and B in X are intersecting if a point xX exists such that (AB)(x) ≠ 0. In this case, A and B intersect at x.

Definition 7 [23]

A fuzzy set A in a fts (X, τ ) is known as the neighborhood of a fuzzy point xλX if and only if Bτ exists such that xλBA; a neighborhood A is known as an open neighborhood if and only if A is fuzzy open. A family comprising all the neighborhoods of xλ is known as a system of neighborhoods of xλ.

Definition 8 [23]

A fuzzy point xα is quasi-coincident with a fuzzy set A, denoted by xαqA, if and only if α + A(x) > 1 or α > (A(x))c.

Definition 9 [23]

A fuzzy set A is quasi-coincident with B and is denoted by AqB if and only if xX exists such that A(x) + B(x) > 1.

It is clear that if A and B are quasi-coincident at x, then both A(x) and B(x) are non-zero at x; hence, A and B intersect at x.

Definition 10 [23]

A fuzzy set A in an fts (X, τ ) is known as a quasi-neighborhood (Q-neighborhood) of xλ if and only if ∃ A1τ, such that A1A and xλqA1. A family Uxλ comprising all the Q-neighborhoods of x is known as the Q-neighborhood system of x. The intersection of two Q-neighborhoods of xλ is a quasi-neighborhood.

Theorem 1 [8]

Let (X, τ1) and (X, τ2) be two fuzzy topological spaces. Consider fuzzy sets τ1(τ2) = {AIX: For any xqA, a τ2-Q-neighborhood A of x exists such that τ1-closure Ā|αA}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Definition 11 [10]

A fts (X, τ ) is known as a fuzzy T0-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ such that xαUco(yβ) or yβVco(xα) in τ.

Definition 12 [10]

A fts (X, τ ) is known as a fuzzy T1-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ such that xαU and yβU, and xαV and yβV.

Definition 13 [10]

A fts (X, τ ) is known as the fuzzy Hausdorff or fuzzy T2-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist such that xαU and yβV, and UV = 0̄.

Definition 14 [10]

A fts (X, τ ) is known as fuzzy regular if and only if for any fuzzy point x and a closed fuzzy set B with xBc, open fuzzy sets U and V exist in τ such that xαU and BV, and U ⊆ 1 − V.

Proposition 1 [25]

fts (X, τ ) is known as a fuzzy regular space if and only if for any fuzzy point xα in IX and Aτ withα < A(x), Bτ exists such that α < B(x) and A.

Definition 15 [22]

A fts (X, τ ) is known as a fuzzy normal space if and only if for each closed set B and an open fuzzy set U in (X, τ ) where BU, Vτ exists such that BVVU.

Lemma 1 [8]

Let τ1 and τ2 be two fuzzy topological spaces on set X. If every τ1-Q-neighbourhood of x is a τ2-Q-neighbourhood of x for all fuzzy points x, then τ1 is coarser than τ2.

Theorem 2 [9]

Let (X, τ1) and (X, τ2) be two fuzzy topological spaces. Consider fuzzy sets τ1(τ2) = {AIX: For any fuzzy set B in X with AqB, a τ2-open set A1 exists such that A1qB and τ1-closure Ā1A}. Subsequently, this family of fuzzy sets will form a fuzzy topology on X, and this topology is known as a mixed fuzzy topology on X.

Theorem 3 [8]

Let τ1 and τ2 be two fuzzy topological spaces on set X. Subsequently, the mixed fuzzy topology τ1(τ2) is coarser than τ2. This is represented as τ1(τ2) ⊆ τ2.

Proof

Because every fuzzy point is also a fuzzy singleton set, the theorem is valid with respect to the definition in Theorem 2.

3. Results

In this section, we analogously define the basic definition of T0, T1, T2 (or Hausdorff), as well as regular and normal mixed fuzzy topological spaces (X, τ1(τ2)), and then provide some results in mixed fuzzy topological spaces. The definitions used herein are as follows:

Definition 16

A mixed fts (X, τ1(τ2)) is known as a fuzzy T0-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ1(τ2) such that xαUco(yβ) or yβVco(xα) in τ1(τ2).

Definition 17

A mixed fts (X, τ1(τ2)) is known as a mixed fuzzy T1-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V in τ1(τ2) exist such that xαU and yβU as well as xαV and yβV.

Definition 18

A mixed fts (X, τ1(τ2)) is known as a fuzzy Hausdorff or fuzzy T2-space if and only if for any pair of fuzzy points xα and yβ in IX with xy, open fuzzy sets U and V exist in τ1(τ2) such that xαU and yβV, and UV = 0̄.

Definition 19

A mixed fts (X, τ1(τ2)) is known as a fuzzy regular space if and only if for any fuzzy point xα in IX and a closed set B in τ1(τ2) with xBc, open fuzzy sets U and V exist in τ1(τ2) such that xαU and BV, and U ⊆ 1 − V.

Definition 20

A mixed fts (X, τ1(τ2)) is known as a fuzzy normal space if and only if for each closed fuzzy set B and an open fuzzy set U in τ1(τ2) where BU, Vτ1(τ2) exists such that BVU.

Example 1

Let us consider the following example of mixed fuzzy topology, which comprises fuzz T0, fuzzy T1 and fuzzy T2.

Let X = {x, y} and xy; therefore

τ1={0¯,1¯,{(x,α),(y,0)},{(x,0),(y,α)},{(x,α),(y,β)}|α,β[0.6,1]},

and

τ2={0¯,1¯,{(x,0.3),(y,0)},{(x,0),(y,0.3)},{(x,0.5),(y,0)},{(x,0),(y,0.5)},{(x,0.3),(y,03)},{(x,0.5),(y,0.5)},{(x,1),(y,0)},{(x,0),(y,1)}}

are fuzzy Hausdorff topological spaces. We constructed τ1(τ2) from these topologies. In addition, τ1(τ2) ⊆ τ2. Therefore we verified the belongingness of {(x, 0.3), (y, 0.3)}, {(x, 0.5), (y, 0.5)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x, 0.5), (y, 0)}, {(x, 0), (y, 0.5)}, {(x, 1), (y, 0)}, and {(x, 0), (y, 1)} in τ1(τ2). As an example, = {(x, 0.3), (y, 0.3)} ∈ τ2 for A1 = {(x, δ): δ > 0.7}; AqA1 and the τ1 closure of (A = {(x, 0.3), (y, 0.3)}) = AA. Because {(x, 0.7), (y, 0.7)} is an open set in τ1, { (x, 0.3), (y, 0.3)} is a closed set in τ1. (Similarly, for A2 = {(y, δ): δ > 0.7} or A3 = {(x, γ), (y, δ) | γ, δ ∈ (0.7, 1]}; AqA2 and AqA3 are also a τ1 closure of (A = {(x, 0.3), (y, 0.3)}) = AA.)

Therefore Aτ1(τ2). However, for B = {(x, 0.5), (y, 0.5)} ∈ τ2 and A2 = {(x, δ): δ > 0.5} BqA2 and the τ1 closure of (B = {(x, 0.5), (y, 0.5)}) = 1̄ ⊄ B.

Therefore, Bτ1(τ2). Similarly, as in the case of A = {(x, 0.3), (y, 0.3)} ∈ τ2, {(x, 1), (y, 0)} and {(x, 0), (y, 1)} will be members of τ1(τ2). Furthermore, τ1 is the closure of {(x, 1), (y, 0)} = {(x, 1), (y, 0)}, as {(x, 0), (y, 1)} is an open set in τ1; therefore, the complement {(x, 1), (y, 0)} is closed. In addition, the τ1 closure of {(x, 0), (y, 1)} = {(x, 0), (y, 1)} as {(x, 1), (y, 0)} is an open set in τ1.

The complement of {(x, 0.3), (y, 0)} in τ1 is {(x, 0.7), (y, 1)}, which is an open set such that {(x, 0.3), (y, 0)} is a closed set in τ1. Therefore, as in the case of A = {(x, 0.3), (y, 0)} ∈ τ1(τ2), we have {(x, 0), (y, 0.3)} ∈ τ1(τ2).

However, the complement of {(x, 0.5), (y, 0)} in τ1 is {(x, 0), (y, 0.5)} and not fuzzy open in τ1; therefore, {(x, 0.5), (y, 0)} is not fuzzy closed, and the τ1-closure of {(x, 0.5), (y, 0)} ⊄ {(x, 0.5), (y, 0)}.

Similarly, {(x, 0.5), (y, 0)}∉ τ1(τ2) and {(x, 0), (y, 0.5)} ∉ τ1(τ2).

Therefore, τ1(τ2) = {0̄, 1̄, {(x, 0.3), (y, 0.3)}, {(x, 0.3), (y, 0)}, {(x, 0), (y, 0.3)}, {(x 1), (y, 0)}, {(x, 0), (y, 1)}}.

Next, we prove that τ1(τ2) is fuzzy T0, fuzzy-T1, and fuzzy-T2.

For xα and yβ in IX, xα ∈ {(x, 1), (y, 0)} ⊆ co(yβ) and yβ ∈ {(x, 0), (y, 1)} ⊆ co(xα).

Additionally, {(x, 1), (y, 0)} ∩ {(x, 0), (y, 1)} = 0̄.

Therefore, τ1(τ2) qualifies as fuzzy-T0, fuzzy-T1, and fuzzy-T2.

Example 2

Let X = {x, y}; subsequently, τ1 = {1̄, 0̄, {(x, 0), (y, 1)}} and τ2 = {1̄, 0̄, {(x, 1), (y, 0)}, {(x, 0.7), (y, 0.7)}, {(x, 0.7), (y, 0)}, {(x, 1), (y, 0.7)}} are fuzzy topologies. Therefore, τ1(τ2) = {1̄, 0̄, {(x, 1), (y, 0)}}, since {(x, 1), (y, 0)} is a closed fuzzy set in τ1. For any fuzzy set B in X with {(x, 1), (y, 0)}qB, we have a τ2-open fuzzy set {(x, 1), (y, 0)} such that {(x, 1), (y, 0)}qB and τ1-closure ({(x,1),(y,0)}¯){(x,1),(y,0)}.

Clearly, τ1(τ2) is mixed fuzzy T0 but not mixed fuzzy T1, as xα ∈ {(x, 1), (y, 0)} ⊆ co(yβ) and ∄ Aτ1(τ2) such that yαAco(xβ).

Next, we consider a mixed fuzzy topological space that is fuzzy T1 but not fuzzy T2.

Example 3

Let X = {x, y, z}; subsequently, τ1 = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}, {(x, 0.7), (y, 0)(z, 0)}, {(x, 0.7), (y, 1)(z, 0)}, {(x, 0.7), (y, 0)(z, 1)}, {(x, 0.7), (y, 1)(z, 1)}} and τ2 = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1)(y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}} are fuzzy topologies.

Therefore, τ1(τ2) = {1̄, 0̄, {(x, 1), (y, 1)(z, 0)}, {(x, 1), (y, 0)(z, 1)}, {(x, 0), (y, 1)(z, 1)}, {(x, 1), (y, 0)(z, 0)}, {(x, 0), (y, 1)(z, 0)}, {(x, 0), (y, 0)(z, 1)}}.

Because every open fuzzy set in τ2 is a closed fuzzy set in τ1. Therefore, every open fuzzy set in τ2 is an open fuzzy set in τ1(τ2), as shown in Example 2.

Next, we demonstrate that for every pair xα, yβ, yα, zβ, or zα, xβ, open fuzzy sets U and V exist in τ1(τ2) such that xαUco(yβ) and yβVco(xα). We will verify this only for the pair xα, yβ; similarly, other cases can be completed. We have open fuzzy sets {(x, 1), (y, 0)(z, 1)} and {(x, 0), (y, 1)(z, 1)} in τ1(τ2) such that xα ∈ {(x, 1), (y, 0)(z, 1)} ⊆ co(yβ) and yβ ∈ {(x, 0), (y, 1)(z, 1)} ⊆ co(xα).

Therefore, τ1(τ2) is fuzzy T1.

However, {(x, 1), (y, 0)(z, 1)} ⊄ co({(x, 0), (y, 1)(z, 1)}).

Therefore, τ1(τ2) is not a mixed fuzzy T2 space.

Next, we establish the following theorems:

Theorem 4

If τ1τ2 and τ1 is fuzzy regular, then τ1τ1(τ2).

Proof

Let A be a τ1-Q-neighborhood of xα, then ∃ A1τ1 such that

xαqA1Aα+A1(x)>1(0<α1)A1(x)>1-α=β(say)A1(x)>β,where 0<β1Bτ1exists such that β<B(x)and τ1-closure of         BA1         (Since τ1is fuzzy regular).

Therefore,

1-α<B(x)1<B(x)+αand τ1-closure B¯A1xαqB,and Bτ2and τ1-closure B¯A1AAτ1(τ2)(By definition of τ1(τ2))Ais a τ1(τ2)quasi-neighborhood of xατ1τ1(τ2).

Hence, the proof is completed.

Theorem 5

If τ1τ2 and τ1 are fuzzy regular, then τ1(τ2) is a fuzzy-T0, fuzzy-T1, and fuzzy-T2 (or Hausdorff) space.

Proof

If τ1 is fuzzy regular, then τ1 is a fuzzy-T0, fuzzy-T1, and fuzzy-T2 (or Hausdorff) space.

Furthermore, τ1τ1(τ2) (by Theorem 4).

1) We prove that τ1(τ2) is a fuzzy T0-space.

Let xα, yβ be any two fuzzy points such that xy. Because τ1 is a fuzzy T0-space, without loss of generality, let an open fuzzy set Uτ1 such that

xαUco(yβ)xαUτ1(τ2)and yβU[since τ1τ1(τ2)]open fuzzy set Uτ1(τ2)such that xαUco(yβ)τ1(τ2)is fuzzy T0space.

2) Next, we prove that τ1(τ2) is a fuzzy T1-space.

Let xα and yβ be any two fuzzy points such that xy. Because τ1 is a fuzzy T1-space, open fuzzy sets V1 and V2 in τ1 exist such that xαV1 and yβV1, and xαV2 and yβV2. Since τ1τ1(τ2), therefore V1 and V2 are in τ1(τ2) such that xαV1 and yβV1, and xαV2 and yβV2. Therefore, τ1(τ2) is a fuzzy T1-pace.

3) Finally, we prove that τ1(τ2) is a fuzzy Hausdorff space.

Let xα and yβ be any two fuzzy points such that xy.

V1 and V2 exist in τ1 such that xαV1 and yβV2 and V1V2 = 0̄ (since τ1 is fuzzy Hausdorff).

V1 and V2 are in τ1(τ2) such that xαV1 and yβV2, and V1V2 = 0̄. [since τ1τ1(τ2)]

τ1(τ2) is a fuzzy Hausdorff space.

Theorem 6

If τ1(τ2) is a fuzzy T0-space, then τ2 is also a fuzzy T0-space.

Proof

We assume τ1(τ2) to be a fuzzy T0-space.

Let xα and yβ be any two fuzzy points such that xy.

Without loss of generality, let an open fuzzy set Uτ1(τ2) exist such that xαUco(yβ).

Since τ1(τ2) ⊆ τ2, Uτ2 such that xαUco(yβ).

Hence, τ2 is a fuzzy T0-space.

Theorem 7

If τ1(τ2) is a fuzzy T1-space, then τ2 is also a fuzzy T1-space.

Proof

Let us assume that τ1(τ2) is a fuzzy T1-space.

Let xα and yβ be any two fuzzy points such that xy. Because τ1(τ2) is a fuzzy T1-space, open fuzzy sets V1 and V2 exist in τ1(τ2) such that xαV1 and yβV1, xαV2, and yβV2. Since τ1(τ2) ⊆ τ2, V1 and V2 exist in τ2 such that xαV1 and yβV1, and xαV2 and yβV2. Hence, τ2 is also a fuzzy T1-space.

Theorem 8

If τ1(τ2) is a fuzzy T2-space (or Hausdorff), then τ2 is also a fuzzy T2-space.

Proof

Let xα and yβ be any two fuzzy points such that xy.

Because τ1(τ2) is fuzzy Hausdorff, V1 and V2 exist in τ1(τ2) such that xαV1 and yβV2, and V1V2 = 0̄.

Hence, V1 and V2 exist in τ2 such thauto-at let xαV1 and yβV2, and V1V2 = 0̄ (since τ1(τ2) ⊆ τ2).

Hence, τ2 is also a fuzzy T2-space.

Theorem 9

Let τ1τ2 and τ2 be fuzzy regular; therefore, τ1(τ2) is also fuzzy regular.

Proof

Let xα be a fuzzy point and C be a closed fuzzy set in τ1(τ2) such that xαCc.

Therefore, C is a closed fuzzy set in τ1(τ2).

⇒ 1 − C is an open fuzzy set in τ1(τ2).

⇒ 1 − C is an open fuzzy set in τ2 (since τ1(τ2) ⊆ τ2).

C is closed in τ2.

⇒open fuzzy sets A1 and A2 exist in τ2 such that xαA1 and CA2 and A1 ⊆ 1 − A2.

Next, for any fuzzy set B such that BqA1

B(z) + A1(z) > 1, for some zX

A1(z) > 1 − B(z)

A1(z) > r, where r = 1− B(z).

The fuzzy regularity of τ2 guarantees that ∃ A2τ2 such that A2(z) > r and A2τ2cl(AtA1, since τ1τ2.

Therefore,

τ1-cl(A2)={B|A2B,where Bis a closed fuzzy set in τ1}{B|A2B,Bis a closed fuzzy set in τ2}=τ2-cl(A2).[Every closed fuzzy set in τ2is a closed fuzzy set in τ1].

Therefore, A2qB and A2τ1cl(A2) ⊆ A1.

Furthermore, A1τ1(τ2), and similarly, we can show that A2τ1(τ2).

⇒ open fuzzy sets A1 and A2 exist in τ1(τ2) such that xαA1 and CA2, and A1 ⊆ 1 − A2.

Hence, (X, τ1(τ2)) is a fuzzy regular space.

Corollary 1

For any non-empty crisp set X, if τ2τ1(τ2), then τ2 = τ1(τ2) because τ1(τ2) ⊆ τ2.

Next, we consider a mixed fuzzy topology comprising fuzzy regular and fuzzy normal.

Example 4

Let X = {x, y} and xy. Let τ1 be the fuzzy topology generated by the base and τ2 be the fuzzy topology generated by the base , where α is a rational number} ∪ {yβ | β ∈ (0, 1] and β is a rational number} ∪ {∅︀}. Therefore, τ1 and τ2 are fuzzy regular and fuzzy normal spaces, respectively. Next, we show that the mixed fuzzy topology τ1(τ2) is the same as the fuzzy topology τ2. We know that τ1(τ2) ⊆ τ2. Moreover, for any element in the base, e.g., B = {xα | α ∈ (0, 1], where α is a rational number}, the fuzzy set C = {(x, 1 − α), (y, 1)} is an open fuzzy set in τ1 and a complement of B in τ1. Therefore B is a closed fuzzy set in τ1, and τ1-closure of B = B. Therefore, for any fuzzy set A with AqBτ2 open fuzzy set B such that AqB and τ1-closure of B = B. Therefore, Bτ1(τ2). Similarly, any base element {yβ | β ∈ (0, 1], where β is a rational number} is also in τ1(τ2); this implies τ2τ1(τ2). Therefore, τ1(τ2) = τ2.

Next, we prove the fuzzy normality and fuzzy regularity of τ1(τ2).

For the fuzzy regularity, let x be any fuzzy point and B be a closed fuzzy set in τ1(τ2) such that

xBcBc(x)λ1-B(x)λB(x)1-λB(x)=1-λor B(x)<1-λ.

If B(x) = 1 − λ, then 1 − λ is a rational number.

B ⊆ {(x, 1 − λ), (y, 1)} = V is an open fuzzy set in τ1(τ2) and xU = {(x, λ), (y, 0)} ∈ τ1(τ2) such that U = {(x, λ), (y, 0)} ⊆ 1 − V = {(x, λ), (y, 0)}. Therefore, in this case, τ1(τ2) is fuzzy regular.

Next, if B(x) < 1−, then ∃ is a rational number δ such that B(x) < δ < 1 − λ.

Therefore, B ⊆ {(x, δ), (y, 1)} = V is an open fuzzy set in τ1(τ2).

Furthermore,

δ<1-λ1-δ>1-(1-λ)1-δ>λis a rational number rsuch that 1-δ>r>λ

⇒ {(x, r), (y, 0)} = U is an open set in τ1(τ2) such that xU and Vc = {(x, 1 − δ), (y, 0)}, resulting in UVc; therefore, τ1(τ2) is fuzzy regular in both cases.

Next, we prove that τ1(τ2) is fuzzy normal.

Let B be any closed fuzzy set and U be an open fuzzy set such that BU.

B(x) ≤ U(x), ∀ xX and U is an open fuzzy set in τ1(τ2). B(x) = 1 − r for some rational number

r(0,1]B(x)B(x)+U(x)2U(x) and B(x)+U(x)2 is a rational number in (0, 1].

Therefore, V(x)=B(x)+U(x)2 and V(y)=B(y)+U(y)2 defines a fuzzy open set in τ1(τ2).

Furthermore, B(x) ≤ V (x) ≤ U(x) ∀ xX. Similarly, 1−V yields rational values for both x and y; therefore, 1−V is open in τ1(τ2) ⇒ V is fuzzy closed as well as ⇒ V (x) = (x) ∀ xX.

B(x)V(x)V¯(x)U(x)xXB(x)V(x)V¯(x)U(x).

Therefore, τ1(τ2) is a normal mixed fuzzy topological space.

Theorem 10

If τ1τ2 and τ2 are fuzzy normal, then τ1(τ2) is also fuzzy normal.

Proof

Since τ2 is fuzzy normal, it is fuzzy regular. Therefore, from Corollary 1, we obtain τ1(τ2) = τ2. Hence, τ1(τ2) is also normal.

Theorem 11

A fuzzy T2 mixed fts (X, τ1(τ2)) is a mixed fuzzy T1-space, and a mixed fuzzy T1 space is a mixed fuzzy T0-space.

Proof

Let (X, τ1(τ2)) be a T2 mixed fuzzy topological space.

Let xα and yβ be two fuzzy points such that xy. Therefore, open fuzzy sets O1 and O2 exist such that xαO1co(yβ), yβO2co(xα), and O1co(O2). Therefore, xαO1co(yβ) and yβO2co(xα); as such, (X, τ1(τ2)) is a T1 mixed fuzzy topological space.

Let (X, τ1(τ2)) be a T1 mixed fuzzy topological space.

Let xα and yβ be two fuzzy points such that xy. Therefore, open fuzzy sets O1 and O2 exist such that xαO1co(yβ) and yβO2co(xα). Consequently, xαO1co(yβ) or yβO2co(xα). Hence, (X, τ1(τ2)) is a mixed fuzzy T0 topological space.

4. Conclusion

In this study, we defined fuzzy-T0, fuzzy-T1, fuzzy-T2 (or Hausdorff), fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. We established the relationships among fuzzy-T0, fuzzy-T1, fuzzy-T2, fuzzy regular, and fuzzy normal spaces in mixed fuzzy topological spaces. Only a few theorems have been established to investigate separation axioms in mixed fuzzy topological spaces. More properties and theorems pertaining to separation axioms will be established in the near future using these established results in the context of mixed fuzzy topological spaces.

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