Fuzzy topology applications

Definition 5 .

Let τ be a family of [3, 2]-fuzzy subsets of a non-empty set X. If

1. 1X, 0X τ where 1X=[1,0] and 0X=[0,1],

2. D1D2 τ, for any D1, D2 τ,

3. iIDi τ, for any {Di}iI τ,

then τ is called a [3, 2]-fuzzy topology on X and [X, τ] is a [3, 2]-fuzzy topological space. We call D an open [3, 2]-FS if it is a member of τ and call its complement a closed [3, 2]-FS.

Remark 2 .

We call τ={1X, 0X} the indiscreet [3, 2]-fuzzy topology on X. If τ contains all [3, 2]-fuzzy subsets, then we call τ the discrete [3, 2]-fuzzy topology on X.

Example 4 .

Let τ={1X, 0X, D1, D2, D3, D4, D5} be the family of [3, 2]-fuzzy subsets of X={x1, x2}, where

Hence, τ is [3, 2]-fuzzy topology on X.

Remark 3 .

We showed that every fuzzy set D on a set X is a [3, 2]-fuzzy set having the form D={r, αD[r], 1 αD[r] :  r X}. Then, every fuzzy topological space [X, τ1] in the sense of Chang is obviously a [3, 2]-fuzzy topological space in the form τ={D :  αD τ1} whenever we identify a fuzzy set in X whose membership function is αD with its counterpart D={r, αD[r], 1 αD[r] :  r X}. Similarly, one can note that every intuitionistic fuzzy topology [Pythagorean fuzzy topology] is [3, 2]-fuzzy topology. The following examples explain this note.

Example 5 .

Consider τ={1X, 0X, D1, D2} as family of fuzzy subsets of X={x}, where

Then, τ is fuzzy topology on X, and hence it is [3, 2]-fuzzy topology.

Example 6 .

Let τ={1X, 0X, D1, D2} be the family of [3, 2]-fuzzy subsets on X={x1, x2} where

Hence, τ is [3, 2]-fuzzy topology. On the other hand, τ is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.

Definition 6 .

Let [X, τ] be a [3, 2]-fuzzy topological space and D={x, αD[x], βD[x] :  x X} be a [3, 2]-FS in X. Then, the [3, 2]-fuzzy interior and [3, 2]-fuzzy closure of D are, respectively, defined by

1. cl[D]={H : H is a closed [3, 2]-FS in X and D H}.

2. int[D]={G : G is an open [3, 2]-FS in X and G D}.

Remark 4 .

Let [X, τ] be a [3, 2]-fuzzy topological space and D be any [3, 2]-FS in X. Then,

1. int[D] is an open [3, 2]-FS.

2. cl[D] is a closed [3, 2]-FS.

3. int[1X]=cl[1X]=1X and int[0X]=cl[0X]=0X.

Example 7 .

Consider the [3, 2]-fuzzy topological space [X, τ] in Example 4. If D={c1, 0.67, 0.81, c2, 0.75, 0.74}, then int[D]=0X and cl[D]=1X.

Theorem 5 .

Let [X, τ] be a [3, 2]-fuzzy topological space and D1, D2 be [3, 2]-FSs in X. Then, the following properties hold:

1. int[D1] D1 and D1 cl[D1].

2. If D1 D2, then int[D1] int[D2] and cl[D1] cl[D2].

3. D1 is an open [3, 2]-FS if and only if D1=int[D1].

4. D1 is a closed [3, 2]-FS if and only if D1=cl[D1].

Proof

[1] and [2] are obvious.

[3] and [4] follow from Definition 6.

Corollary 1 .

Let [X, τ] be a [3, 2]-fuzzy topological space and D1, D2 be [3, 2]-FSs in X. Then, the following properties hold:

1. int[D1] int[D2] int[D1 D2].

2. cl[D1D2] cl[D1]cl[D2].

3. int[D1D2]=int[D1]int[D2].

4. cl[D1] cl[D2]=cl[D1 D2].

Proof

[1] and [2] follows from [1] of the above theorem.

[3]: since int[D1D2] int[D1] and int[D1D2] int[D2], we obtain int[D1D2] int[D1]int[D2]. On the other hand, from the facts int[D1] D1 and int[D2] D2, we have int[D1]int[D2] D1D2 and int[D1]int[D2] τ; we see that int[D1]int[D2] int[D1D2], and hence int[D1D2]=int[D1]int[D2].

[4] can be proved similar to [3].

Theorem 6 .

Let [X, τ] be a [3, 2]-fuzzy topological space and D be [3, 2]-FS in X. Then, the following properties hold:

1. cl[Dc]=int[D]c.

2. int[Dc]=cl[D]c.

3. cl[Dc]c=int[D].

4. int[Dc]c=cl[D].

Proof

We only prove [1]; the other parts can be proved similarly.

Let D={x, αD[x], βD[x] :  x X} and suppose that the family of open [3, 2]-fuzzy sets contained in D is indexed by the family {x, αUi[x], βUi[x] :  i J}. Then, int[D]={x, αUi[x], βUi[x]}. Therefore, int[D]c={x, βUi[x], αUi[x]}. Now, Dc={x, βD[x], αD[x]} such that αUi αD, βUi βD for each i J. This implies that {x, βUi[x], αUi[x] :  i J} is the family of all closed [3, 2]-fuzzy sets containing Dc. That is, cl[Dc]={x, βUi[x], αUi[x]}. Hence, cl[Dc]=int[D]c.

Definition 7 .

Let f : X⟶Y be a map and A and B be [3, 2]-fuzzy subsets of X and Y, respectively. The functions of membership and non-membership of the image of A, denoted by f[A], are, respectively, calculated by

The functions of membership and non-membership of preimage of B, denoted by f1[B], are, respectively, calculated by

Remark 5 .

To show that f[A] and f1[B] are [3, 2]-fuzzy subsets, consider γA[z]]5=αA[z]]3+[βA[z]]2. If f1[y] is non-empty, then we obtain

In contrast, f1[y]=ϕ leads to the fact that [αf[A][y]]3+[βf[A][y]]2=1.

It is easy to prove the case of f1[B].

Theorem 7 .

Let f : X⟶Y be a map s.t. A and B are [3, 2]-fuzzy subsets of X and Y, respectively. Then, we have

1. f1[Bc]=f1[B]c.

2. f[A]cf[Ac].

3. If B1B2, then f1[B1]f1[B2] where B1 and B2 are [3, 2]-fuzzy subsets of Y.

4. If A1A2, then f[A1]f[A2] where A1 and A2 are [3, 2]-fuzzy subsets of X.

5. f[f1[B]]B.

6. Af1[f[A]].

Proof

• [1]

  Consider v X and let B be a [3, 2]-fuzzy subset of Y. Then,

  αf1Bcv=αBcfv=βBfv=βf1Bv=αf1Bcv.

  [17]

• Similarly, one can have βf1[Bc][v]=βf1[B]c[v]. Therefore, f1[Bc]=f1[B]c, as required.

• [2]

  For any w Y such that f1[w] ϕ and for any [3, 2]-fuzzy subset A of X, we can write

  γfAw5=αfAw3+βfAw2=supzf1wαAz3+infzf1wβAz2=supzf1wγAz5βAz2+infzf1wβAz2supzf1wγAz5infzf1wβAz2+infzf1wβAz2=supzf1wγAz5.

  [18]

• Now from [18], we have

  αfAcw=supzf1wαAcz=supzf1wβAz=supzf1wγAz5αAz3supzf1wγAz5supzf1wαAz3γfAw5αfAw3=βfAw=αfAcw.

  [19]

• The proof is easy when f1[w]=ϕ. Following a similar technique, we obtain βf[Ac][w] βf[A]c[w], which means that f[A]cf[Ac].

• [3]

  Assume that B1B2. Then, for each v X, αf1[B1][v]=αB1[f[v]] αB2[f[v]]=αf1[B2][v]. Also, βf1[B1][v] βf1[B2][v]. Hence, we obtain the desired result.

• [4]

  Assume that A1A2 and w Y. The proof is easy when f[w]=ϕ. So, presume that f[w] ϕ. Then,

  αfA1w=supzf1wαA1zsupzf1wαA2z=αfA2w.

  [20]

• Thus, αf[A1] αf[A2] follows. Similarly, we have βf[A1] βf[A2].

• [5]

  For any w Y s.t. f[w] ϕ, we find that

  αff1Bw=supzf1wαf1Bz=supzf1wαBfzαBw.

  [21]

• On the other hand, we have αf[f1[B]][w]=0 αB[w] when f[w]=ϕ. Similarly, we have βf[f1[B]][w]=0 βB[w].

• [6]

  For any v X, we have

  αf1fAv=αfAfv=supzf1fvαAzαAv.

  [22]

• Similarly, we have βf1[f[A]] βA.

Theorem 8 .

Let X and Y be two non-empty sets and f : X⟶Y be a map. Then, the following statements are true:

1. f[iIAi]=iIf[Ai] for any [3, 2]-fuzzy subset Ai of X.

2. f1[iIBi]=iIf1[Bi] for any [3, 2]-fuzzy subset Bi of Y.

3. f[A1A2] f[A1]f[A2] for any two [3, 2]-fuzzy subsets A1 and A2 of X.

4. f1[iIBi]=iIf1[Bi] for any [3, 2]-fuzzy subset Bi of Y.

Definition 8 .

In a [3, 2]-fuzzy topological space, consider that A and U are two [3, 2]-fuzzy subsets. We call U a neighborhood of A, briefly nbd, if there exists an open [3, 2]-fuzzy subset E such that AEU.

Theorem 9 .

A [3, 2]-fuzzy subset A is open iff it contains a nbd of its each subset.

Proof

The proof is easy.

Definition 9 .

A map f : [X, τ1]⟶[Y, τ2] is said to be [3, 2]-fuzzy continuous if for any [3, 2]-fuzzy subset A of X and for any nbd V of f[A] there is a nbd U of A s.t. f[U]V.

Theorem 10 .

The following statements are equivalent for a map f : [X, τ1]⟶[Y, τ2]:

1. f is [3, 2]-fuzzy continuous.

2. For each [3, 2]-FS A of X and each nbd V of f[A], there is a nbd U of A s.t. for each BU, we obtain f[B]V.

3. For each [3, 2]-FS A of X and for each nbd V of f[A], there is a nbd U of A s.t. Uf1[V].

4. For each [3, 2]-FS A of X and for each nbd V of f[A], f1[V] is a nbd of A.

Proof

•   [1][2]: let f be a [3, 2]-fuzzy continuous map. Consider A as a [3, 2]-FS of X and V as a nbd of f[A]. Then, there is a nbd U of A s.t. f[U]V. If BU, we obtain f[B]f[U]V.

•   [2][3]: assume A as a [3, 2]-FS of X and V as a nbd of f[A]. According to [2], there is a nbd U of A s.t. for each BU, we find f[B]V. Therefore, Bf1[f[B]]f1[V]. Since B is chosen arbitrarily, Uf1[V].

•   [3][4]: presume A is a [3, 2]-FS of X and V is a nbd of f[A]. According to [3], there is a nbd U of A s.t. Uf1[V]. Since U is a nbd of A, there is an open [3, 2]-FS K of X s.t. AKU. On the other hand, we obtain AKf1[V] because Uf1[V]. This means that f1[V] is a nbd of A.

•   [4][1]: suppose that A is a [3, 2]-FS of X and V is a nbd of f[A]. By hypothesis, f1[V] is a nbd of A. So, there is an open [3, 2]-FS K of X s.t. AKf1[V] which means f[K]f[f1[V]]V. Moreover, K is an open [3, 2]-FS, so it is a nbd of A. Hence, we obtain the proof that f is [3, 2]-fuzzy continuous.

Theorem 11 .

A map f : [X, τ1]⟶[Y, τ2] is [3, 2]-fuzzy continuous iff f1[B] is an open [3, 2]-FS of X for each open [3, 2]-FS B of Y.

Proof

Necessity: presume f as a [3, 2]-fuzzy continuous map. Consider an open [3, 2]-FS B of Y s.t. Af1[B]. This directly gives that f[A]B. It follows from Theorem 9 that there is a nbd V of f[A] satisfying VB. Now, f is [3, 2]-fuzzy continuous, so by [4] of Theorem 10, we obtain that f1[V] is a nbd of A. Also, it follows from [3] of Theorem 7 that f1[V]f1[B]. So, f1[B] is a nbd of A. Since A is an arbitrary subset of f1[B], then by Theorem 9, the [3, 2]-FS f1[B] is open.

3.2.1. Sufficiency

Presume A is a [3, 2]-FS of X and V is a nbd of f[A]. Then, τ2 contains a [3, 2]-FS L of s.t. f[A]LV. By hypothesis, f1[L] is an open [3, 2]-FS. Also, we have Af1[f[A]]f1[L]f1[V]. Thus, f1[V] is a nbd of A which demonstrates that f is [3, 2]-fuzzy continuous.

We build the following two examples such that the first one provides a [3, 2]-fuzzy continuous map, whereas the second one presents a fuzzy map that is not [3, 2]-fuzzy continuous.

Definition 10 .

Let X and x X be a fixed element in X. Suppose that r1 [0,1] and r2 [0,1] are two fixed real numbers such that r13+r22 1. Then, a [3, 2]-fuzzy point p[r1, r2]x={x, αp[x], βp[x]} is defined to be a [3, 2]-fuzzy set of X as follows.

for y X. In this case, x is called the support of p[r1, r2]x. A [3, 2]-fuzzy point p[r1, r2]x is said to belong to a [3, 2]-fuzzy set D={x, αD[x], βD[x]} of X denoted by p[r1, r2]x D if r1 αD[x] and r2 βD[x]. Two [3, 2]-fuzzy points are said to be distinct if their supports are distinct.

Remark 6 .

Let D1={x, αD1[x], βD1[x]} and D2={x, αD2[x], βD2[x]} be two [3, 2]-fuzzy sets of X. Then, D1D2 if and only if p[r1, r2]x D1 implies p[r1, r2]x D2 for any [3, 2]-fuzzy point p[r1, r2]x in X.

Definition 11 .

Let r1, r3 [0,1], r2, r4 [0,1], and x, y X. A [3, 2]-fuzzy topological space [X, τ] is said to be

• [1]

  T0 if for each pair of distinct [3, 2]-fuzzy points p[r1, r2]x, p[r3, r4]y in X, there exist two open [3, 2]-fuzzy sets L and K such that

  L=x,1,0,y,0,1,or K=x,0,1,y,1,0.

  [32]
• [2]

  T1 if for each pair of distinct [3, 2]-fuzzy points p[r1, r2]x, p[r3, r4]y in X, there exist two open [3, 2]-fuzzy sets L and K such that

  L=x,1,0,y,0,1,K=x,0,1,y,1,0.

  [33]

Proposition 1 .

Let [X, τ] be a [3, 2]-fuzzy topological space. If [X, τ] is T1, then [X, τ] is T0.

Proof

The proof is straightforward from Definition 11.

Example 10 .

Consider X={c1, c2} with the [3, 2]-fuzzy topology τ={1X, 0X, D}, where D={c1, 1,0, c2, 0,1}. Then, [X, τ] is T0 but not T1 because there does not exist an open [3, 2]-fuzzy set K such that K={x, 0,1, y, 1,0}.

Theorem 14 .

Let [X, τ] be a [3, 2]-fuzzy topological space, r1, r3 [0,1], and r2, r4 [0,1]. If [X, τ] is T0, then for each pair of distinct [3, 2]-fuzzy points p[r1, r2]x, p[r3, r4]y of X, cl[p[r1, r2]x] cl[p[r3, r4]y].

Proof

Let [X, τ] be T0 and p[r1, r2]x, p[r3, r4]y be any two distinct [3, 2]-fuzzy points of X. Then, there exist two open [3, 2]-fuzzy sets L and K such that

Let L={x, 1,0, y, 0,1} exist. Then, Lc={x, 0,1, y, 1,0} is a closed [3, 2]-fuzzy set which does not contain p[r1, r2]x but contains p[r3, r4]y. Since cl[p[r3, r4]y] is the smallest closed [3, 2]-fuzzy set containing p[r3, r4]y, then cl[p[r3, r4]y]Lc, and therefore p[r1, r2]x cl[p[r3, r4]y]. Consequently, cl[p[r1, r2]x] cl[p[r3, r4]y].

Theorem 15 .

Let [X, τ] be a [3, 2]-fuzzy topological space. If p[1,0]x is closed [3, 2]-fuzzy set for every x X, then, [X, τ] is T1.

Proof

Suppose p[1,0]x is a closed [3, 2]-fuzzy set for every x X. Let p[r1, r2]x, p[r3, r4]y be any two distinct [3, 2]-fuzzy points of X; then, x y implies that p[1,0]xc and p[1,0]yc are two open [3, 2]-fuzzy sets such that

Thus, [X, τ] is T1.

The composition R o U.

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