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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 296-302

Published online September 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.3.296

© The Korean Institute of Intelligent Systems

## Door Spaces in Intuitionistic Fuzzy Topological Spaces

AbdulGawad. A. Q. Al-Qubati1,2 and Mohamed El Sayed2

1Department of Mathematics, Hodeidah University, Hodeidah, Yemen
2Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia

Correspondence to :
A. Q. Al-Qubati (gawad196999@yahoo.com)

Received: September 6, 2021; Revised: April 19, 2022; Accepted: September 7, 2022

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.

In this study, a new concept of generalized intuitionistic fuzzy topological space, called intuitionistic fuzzy b~-door space, is introduced and several characterizations of intuitionistic fuzzy b~-door spaces are analyzed. Many examples were introduced to prove the validity of these concepts. Moreover, certain properties and relationships between intuitionistic fuzzy b~-door spaces and other intuitionistic fuzzy topological spaces were investigated.

Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy door space, Intuitionistic fuzzy b~-door space

The notion of an intuitionistic fuzzy set was first defined and further developed by Atanassov [1,2], as a generalization of the fuzzy set proposed by Zadeh [3]. Using the notion of intuitionistic fuzzy sets, Coker [4] introduced intuitionistic fuzzy topological spaces as a generalization of the fuzzy topological spaces proposed by Chang [5]. Several concepts of the fuzzy topological space have been recently extended to intuitionistic fuzzy topological spaces [69]. Parameswari and Thangavelu [10] introduced the concept of b#-open set in ordinary topological spaces and studied several properties of this concept as well as the relationship between the b#-set and other sets. The concept of door spaces was studied in classical topology [11] and then by Donchev [12]. Anjalmose and Thangaraj [13] generalized the concept of door space to fuzzy door spaces.

The aim of this study is to generalize the concept of open and closed sets, called b#-open and b#-closed sets, to intuitionistic fuzzy sets and use them to generalize the concept of topological door spaces to intuitionistic fuzzy topological door spaces. First, we define the concept of intuitionistic fuzzy b~-open and use it to introduce the concept of intuitionistic fuzzy b~-door space. In addition, we introduce the concepts of the intuitionistic fuzzy b~-Baire space and intuitionistic fuzzy b~-submaximal space, as well as their important properties.

Finally, we identify and discuss the relationship between the intuitionistic fuzzy b~-door space and other intuitionistic fuzzy bs-spaces. Furthermore, we prove that the IF b~-door space is a topological property, and that a quasi-compact image of an IF b~-door space is an IF b~-door space.

Section 1 introduces the relative topic, and the basic concepts of the intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces are presented in Section 2. The new concepts for intuitionistic fuzzy b~-door spaces are proposed in Section 3, which is followed by a presentation of the intuitionistic fuzzy functions in b~-door spaces in Section 4. Section 5 presents the conclusion, which briefly summarizes the results and highlights the scope of future research.

### Definition 2.1 ([14])

Let T be a non-empty fixed set. An intuitionistic fuzzy set (IFS) A in T is an object having the form A = {⟨t, μA(t), γA(t)⟩: tT}, where the functions μA:TI and γA : TI denote the degree of membership (μA(t)) and the degree of non-membership (γA(t)) of each element tT to set A, respectively, and 0 ≤ μA(t) + γA(t) ≤ 1, for each tT.

### Definition 2.2 ([14])

Let A and B be IF sets of the form A = {⟨t, μA(t), γA(t)⟩: tT} and B = {⟨t, μB(t), γB(t)⟩: tT}. Then

• (I) AB if μA(t) ≤ μB(t) and νA(t) ≥ νB(t), ∀ tT.

• (II)
• A = B if AB and BA.

(III) Ac = {⟨t, νA(t), μA(t)⟩: tT}.

• (IV) AB = {⟨t, μA(t) ∧ μB(t), νA(t) ∨ νB(t)⟩: tT}.

• (V) AB = {⟨t, μA(t) ∨ μB(t), νA(t) ∧ νB(t)⟩: tT}.

• (VI) 0 = {⟨t, 0, 1⟩: tT}, and 1 = {⟨t, 1, 0⟩: tT}.

• Further details regarding the operations of the IF-sets, IF points, IF functions, and other concepts used in this study can be found in previous references [69,15].

### Definition 2.3 ([4])

An intuitionistic fuzzy topology (IFT) on a non-empty set T is a family Ψ of IF in T that satisfy the following axioms:

• (I) 0, 1ψ.

• (II) If G1, G2ψ, then G1G2ψ.

• (III) If Gψ for each λ ∈ Λ, then ⋃ λ∈ΛGλψ.

• In this case, pair (T, ψ) is called an intuitionistic fuzzy topological space (IFTS) denoted by T, and each intuitionistic fuzzy set in Ψ is an intuitionistic fuzzy open set (IFOS) of T. The complement Ac of IFOS A in IFTS (T, ψ) is the IF closed set (IFCS) in T.

In this study, we introduce the concept of IF b~-open and closed sets as follows:

### Definition 2.4

An IFS A of an IFTS (T, ψ) is an

• (i) intuitionistic fuzzy b~-open set (IF b~OS) if

A=(cl(int(A)(int(cl(A))),and

• (ii) intuitionistic fuzzy b~-closed set (IF b~CS) if

A=(cl(int(A))(int(cl(A))).

### Example 2.5

Let T = {a, b} and M be an IF set on T defined as follows:

M=t,(a/0.5,b/0.5),(a/05,b/0.5).

Then the family ψ = {0, 1, M} is an IFTS on T

IntM=M,Mc=Mand cl(int(M))=M.

Also, cl(M) = M and int(cl(M)) = M.

Therefore, M = (cl(int(M)) ∪ (int(cl(M))), thus M is intuitionistic fuzzy b~-open.

### Example 2.6

Let T = {a, b, c} and M and N be IF sets on T, which are defined as follows:

M=t,(a/1.,b/0.0,c/0.5),(a/0.0,b/1.0,c/0.5),N=t,(a/0.0,b/0.0,c/0.5),(a/1.0,b/1.0,c/0.5).

Then the family ψ = {0, 1, M, N} is an IFT on T.

IntM=M,   and   cl(int(M))=Nc.

Also, cl(M) = Nc and int(cl(M)) = int(Nc) = M.

Thus, (cl(int(M))∪(int(cl(M))) = Nc and cl(int(M))∩ (int(cl(M))) = M; hence, M is not IF b~-open, but is IF b~-closed.

### Example 2.7

Let T = {a, b} and N be an IF set on T defined as follows:

N=t,(a/0.4,b/0.3),(a/0.6,b/0.7).

Then the family ψ = {0, 1, N} is an IFT on T.

Let K = ⟨t, (a/0.6, b/0.7), (a/0.3, b/0.2)⟩ be an IFS on T.

Clearly, (cl(int(K)) ∩ (int(cl(K))) = ⟨t, (a/0.6b/0.7), (a/0.4, b/0.3)⟩ ⊂ K.

Then, K is not IF b~-closed.

### Definition 2.8

Let A be any IF b~-set in IFTS T. Then, the intuitionistic fuzzy b~-closure (IF b~-closure) and intuitionistic fuzzy b~-interior (IF b~interior) of A are defined as follows:

IF b-cl(A)={F:AF,Fis IF bCS in T}.IF b-int(A)={U:UA,Uis IF bOS in T}.

### Definition 2.9 ([7])

Let T be an IFTS and let MT the intuitionistic fuzzy collection.

ψM = {OM : Oτ} is an IFT on T, and pair (M, ψM) is called the IF subspace of an IFTS T.

### Definition 2.10

An IF-point P in an IFTS T is an IF-isolated point if {P} is an IFOS.

### Definition 3.1

An IFTS T is an intuitionistic fuzzy b~-door space (briefly IF b~-door space) if every IF b~-set in T is either IF b~-open or IF b~-closed.

### Example 3.2

Let T = {a, b} and M be an IF set on T, which is defined as follows:

M=t,(a/0.8,b/0.7),(a/0.2,b/0.3).

Then, family ψ = {0, 1, M} is an IFTS on T. The IF set M in T is the IF O set and the complement of an IF setM in T is the IF C set; therefore, the IF sets in T are either IF O or IF C. Therefore, (T, ψ) is an IF door space.

### Example 3.3

Let T = {a, b} and M be an IF set on T defined as follows:

M=t,(a/0.5,b/0.5),(a/0.5,b/0.5).

Then, family ψ = {0, 1, M} is an IFT on T. The IF set M in T is the IF b~O set and the complement of an IF b~-set M in T is the IF b~C-set; therefore, the IF b~-sets in T are either IF b~O or IF b~C. Therefore, (T, ψ) is an IF b~ door space.

### Definition 3.4

Let T be an IFTS, and let A be an IFS in T Then, A is called an IF-nowhere dense set if no non-zero IF-open set B in T exists such that BIF − cl(A). That is,

IF-int(cl(A))=0T˜.

### Example 3.5

Let T = {a, b, c} and M, N, and C be IF sets on T, which are defined as follows:

M=t,(a/0.6,b/0.6,c/0.6),(a/0.3,b/0.3,c/0.4),N=t,(a/0.5,b/0.5,c/0.5),(a/0.4,b/0.4,c/0.4),C=t,(a/0.3,b/0.3,c/0.4),(a/0.5,b/0.5,c/0.4).

Then, the family ψ = {0, 1, M} is an IFT on T.

IF-int(cl(Mc))=0T˜,IF-int(cl(C))=0T˜,IF-int(cl(N))=1T˜,and IF-int(cl(M))=1T˜.

Then, Mc and C are IF-nowhere dense, but M and N are not IF-nowhere dense sets in (T, ψ).

For IF b~-sets, we introduce the definition of IF b~-nowhere dense sets as follows:

### Definition 3.6

Let T be an IFTS, and A be an IFS in T. Then, A is called an IF b~-nowhere dense set if no non-zero IF b~-open set B in T exists such that BIFb-cl(A). That is,

IFb-int(b-cl(A))=0T˜.

### Definition 3.7

Let T be an IFTS. Then, T is called the IF Baire space if IF-int(⋃λ∈ΛAλ) = 0, where A is an IF-nowhere dense set in T.

### Example 3.8

In Example 3.5, Mc and C are IF-nowhere dense sets in (T, ψ). Also, IF − int(McC) = 0. Therefore, (T, ψ) is an IF-Baire space.

For IF b~-sets, we introduce the definition of the IF b~-Baire space as follows:

### Definition 3.9

Let T be an IFTS. Then, T is called the IF b~ Baire space if IFb~-int(⋃λ∈ΛAλ) = 0, where A is an IF b~-nowhere dense set in T.

### Remark 3.10

Example 3.8 will be useful in determining whether the IF sets in Example 3.5 should be replaced the by the IF b~-sets, obtaining the IF b~-nowhere dense set and then the IF b~ Baire space.

### Definition 3.11

Let T be an IFTS. Then, T is an IF b~-submaximal space if every IF b~-dense set is an IF b~-open set in T.

### Definition 3.12

An IF b~S A in an IFTS T is called the IF b~ first category if (i=1Ai)=A, where Ai indicates the IF b~-nowhere dense sets in T. An IF b~S, which is not of the IF b~-first category, is of the IF b~ second category.

### Definition 3.13

An IFTS T is called the IF b~-first-category space (i=1Ai)=1T˜, where Ai indicates the IF b~-nowhere dense sets in T. If T is not an IF b~-first category, then T is said to be an IF b~-second category space.

### Theorem 3.14

Every IF b~-door space T is an IF b~-submax-imal space.

Proof

Let A ⊆ 1 be an IF b~-dense set in T. If A is not IF b~O, then A is an IF b~C because T is an IF b~-door space. Then, A = Ā = 1 and A is an IF b~O (IF b~C). Thus, T is IF b~-submaximal.

The converse of the aforementioned theorem does not need to be true because not all IF b~-open sets are dense, except 1.

### Theorem 3.15

If A is an IF b~-closed set in T with IFb~-int(A) = 0, then A is an IFb~-nowhere dense set in T.

Proof

Let A be an IF b~-closed set in T with IFb~-int(A) = 0 in T. Then, IFb~-cl(A) = A and IFb~-int(A) =; hence IFb~-int(b~-cl(A)) = IFb~-int(A) = 0 in T. Therefore, A is an IFb~-nowhere dense set in T.

The converse of the aforementioned theorem does not need to be true in general; as shown in Example 3.5, C is an IF-nowhere dense set that is not an IF-closed set, as well as for the IF b~-set.

### Theorem 3.16

Every IF subspace of an IF b~ door space (T, ψ) is an IF b~-door space.

Proof

Let (Y, ψY) be a subspace of an IF b~-door space (T, ψ), then YT. Let AY; since (T, ψ) is an IF b~-door space, then A is either IF b~-open or IF b~-closed in (T, ψ), and hence in (Y, ψY). Thus, (Y, ψY) is an IF b~-door space.

### Definition 3.17

An IFTS T is called an IF b~ D-Baire space if every IF b~-first category set in T is an IF b~-nowhere dense set. That is, T is an IF b~ D-Baire space if IFb~-int(cl(A)) = 0 for each IF b~-first category set A in T.

### Proposition 3.18

If T is an IF b~ D-Baire space, then T is an IF b~ Baire space.

Proof

Let A be an IF b~ first-category set in an IF b~ D-Baire space T. Then, A=(i=1Ai)=0T˜, where Ai indicates IF b~-nowhere dense sets and A is an IF b~-nowhere dense set in T. Then, we have IF b~-int(cl(A)) = 0, and IF b~-int(A) ≤ IFb~-intcl(A) implies that IF b~-int(A) = 0. Hence, IFb-int(i=1Ai)=0T˜, where Ai indicates IF b~ nowhere dense sets in T. Therefore, T is an IF b~-Baire space.

### Proposition 3.19

Let T be an IF b~-first category space. Then, T is not an IF b~D-Baire space.

Proof

Let T be an IF b~-first-category space. Then, (i=1Ai)=1T˜, where Ai indicates IF b~-nowhere dense sets in T. For the IF b~-first category set 1, we have IFb~-intcl(1) = 0. Hence, T is not an IF b~ D-Baire space.

### Definition 3.20

An IFTS (T, ψ) is an IF b~-Hausdorff if every IF two disjoint points can be separated by IF b~-disjoint open sets.

### Lemma 3.21

Let P = t(α,β) be an IFP in (T, ψ); then, {P} is an IF b~ O only if {P} is an IFOS.

### Lemma 3.22

Let U and V be two IF b~-sets in (T, ψ). If U is an IF b~O and V is an IFO, then UV is an IF b~O in (T, ψ).

### Theorem 3.23

An IF b~-Hausdorff door space T has at most one limit point.

Proof

Let P = t(α, β) and q = y(γ, ρ) be two IF points in T. Because T is an IF b~-Hausdorff, two IF b~-open sets exist, U and V, such that PU, qV, and UV = 0. Because (T, ψ) is an IF b~-door space, set A = (U \ P) ∪ q is either IF b~O or an IF b~C. Then, in the first case by Lemma 3.21, AV = {q} is IF b~-open and by Lemma 3.22 it is IF b~O. In the second case, Ac is IF b~OS; thus, AcU = {P} is IF b~O, and hence IFO. Therefore, in both cases, at least one of the two IF points is an isolated IF point in T, and by contradiction, the request is proved.

### Definition 3.24

An IFTS (T, ψ) is an IF b~-T1/2 space if every IF point {p} is either an IF b~-open set or IF b~ closed set in T.

### Theorem 3.25

Every IFTS (T, ψ) is an IF b~-T1/2 space.

Proof

Let P be an IF point in T. By Propositions 4.3 and 4.4 of [10], {P} is either IF b~-open or IF-nowhere dense because every IF-nowhere dense set is IF b~ CS. Then, {P} is either IF b~ OS or IF b~ CS. Thus, T is an IF b~-T1/2 space.

### Theorem 3.26

Every IF door space (T, ψ) is an IF b~-space.

Proof

Clearly.

The converse of the aforementioned theorem does not need to be true in general because not all IF bs-OS or IF bs-CS are IFO or IFC, respectively; see Example 2.6.

### Theorem 3.27

Let (T, ψ) be an IFTS; then, the following are equivalents.

• (i) (T, ψ) is an IF-discrete space.

• (ii) (T, ψ) is an IF b~-door space.

Proof

(i) ⇒ (ii) Suppose that (T, ψ) is an IF discrete; then, every IF set is an IFO or IFC, and then IF b~-open or IF b~-closed. Thus, (T, ψ) is an IF b~-door space.

(ii) ⇒ (i) Let P = x(α,β) be an IFP in T. Then, if {P} is not IF b~-open, then {P} is IF b~-closed and {P} = (IFb~-cl(int(P)) ∩ (IFb~-int(cl(P))). Because IFb~int({P}) is not empty and {P} = (IFb~-cl(P)), then {P} = (IFb~-cl(int(P))∩(IFb~-int(p)) = IFb~-int({P}). In both cases, {P} is an IFO. Therefore, (T, ψ) is an IF discrete space.

### Definition 3.28

An IF b~-TS (T, ψ) is irreducible if every IF b~ set in (T, ψ) is IF b~ connected, equivalent to every non-void IF b~ OS in (T, ψ) being dense.

The following theorem introduces an additional condition for the converse of Theorem 3.14 to be true.

### Theorem 3.29

Every IF irreducible submaximal space (T, ψ) is an IF b~-door space.

Proof

Let A be an IF b~-set in (T, ψ). If A is IF b~-dense, then because (T, ψ) is IF b~-submaximal, A is IF b~-open; if A is not IF b~-dense, then we can find a non-void IF b~O set BAc. Because (T, ψ) is irreducible, B and Ac are IF b~-dense. In addition, because (T, ψ) is IF b~-submaximal, Ac is IF b~ O or equivalently, A is IF b~ C. Thus, in any case, A is either IF b~ O or IF b~ C. Therefore, (T, ψ) is an IF b~-door space.

### Definition 3.30

Let (T, ψ) be an IFTS. An intuitionistic fuzzy family I in T is called an intuitionistic fuzzy ideal (IFI) if it is hereditary and has additivity that satisfies the following properties:

• (i) A ∈ IFI and BA implies B ∈ IFIA (hereditary),

• (ii) A ∈ IFI and B ∈ IFI, then AB ∈ IFI (finite additivity).

Let us denote Icd as the IFI of the intuitionistic fuzzy closed discrete sets in T.

### Definition 3.31

An IFTS topological space (T, ψ, I) is called an IFI b~-door space if every IF-set in T is either IF b~-open, IF b~-closed, or belongs to IFI.

### Remark 3.32

Clearly, every intuitionistic fuzzy door space is an intuitionistic fuzzy IFI b~-door space.

### Definition 3.33

An IFTS T is called IF b~-extremally disconnected if the IF b~-closure of an IF b~-open set is IF b~-open.

### Proposition 3.34

Let T be an IFTS. Then, the following conditions are equivalent:

• (i) T is an IF b~-extremally disconnected b~-door space.

• (ii) Every IFS A in T is either IF b~-open, or both IF b~-closed and IF discrete.

Proof

(i) ⇒ (ii) First, IFS A belongs to Icd only if A has no IF limit point. Suppose that A is not IF b~ OS and does not belong to d(A) ≠ ∅. Because T is an IF b~-door space, A is IF b~-closed. Let td(A) ⊆ A, and S = Ac ∪ {p}. Because P is an IF limit point of A, S cannot be IF b~-open; for in that case, S would be an IF b~-open neighborhood of P containing a point of A different than P. Clearly, this is impossible because SA = {P}. Because S is not IF b~-open and because T is an IF b~-door space, S is IF b~-closed. Thus, Ac ∪ {p} = IF-cl(Ac) ∪ IF-cl{p}; hence, AcIF-cl(Ac) ⊆ Ac ∪ {p}. By assumption, because T is IF b~-extremally disconnected, IF-cl(Ac) is IF b~-open. Because S is IF b~-closed, AcIF-cl(Ac) ⊆ (Ac) ∪ {p} = S. Thus, Ac = IF-cl(Ac) or equivalently, A is IF b~-open. Thus, this contradicts with the assumption that A is not IF b~-open.

(ii) ⇒ (i) Let T be an IF b~-door space. We must demonstrate that T is IF b~-extremally disconnected. Let AT be IF b~-open. Then, IF-cl(A) = Ad(A)). If d(A) = ∅, then I-cl(A) = A is IF b~-open. If d(A) ≠ ∅, then d(IF-cl(A)) ≠ ∅ and IF-cl(A) are IF b~-open, which follows (ii).

### Definition 4.1

Let f : (T, ψ) → (Y, δ) be an IF function. Then, f is IF b~-quasi compact if A ⊆ 1 is IF b~ O in (T, ψ) such that f−1(f(A)) = A. Then, f(A) is IF b~ O in (Y, δ).

### Theorem 4.2

An IF b~-door space has a topological property.

Proof

Let f : (T, ψ) → (Y, δ) be an IF b~-homemorphism from an IF b~-door space (T, ψ) into another IF b~-TS (Y, δ). Let A be IF b~S in (Y, δ), that is, AY. Then, f−1(A) is IF b~S in (T, ψ), that is, f−1(A) ⊆ T. Because (T, ψ) is an IF b~-door space, f−1(A) is IF b~O or an IF b~CS in (T, ψ), because f(f−1(A)) = A. Then, A is either IF b~OS or IF b~CS in (Y, δ). Thus, (Y, δ) is the IF b~-door space.

### Theorem 4.3

An IF-quasi-compact image of an IF b~-door space is an IF b~-door space.

Proof

Let f : (T, ψ) → (Y, δ) be an IF-quasi compact function from an IF b~-door space (T, ψ) into another IF b~-TS (Y, δ). Let B be IF b~S in (Y, δ). We must prove that B is either IF b~O or IF b~C in (Y, δ). Because (T, ψ) is an IF b~-door space and f−1(B) = A, then A is either IF b~O or IF b~C in (T, ψ). Suppose that A is IF b~O, and clearly f(A) ⊆ B. Thus, f−1f(A) ⊆ f−1(B) = Af−1f(A) or equivalently, A = f−1f(A). By assumption, f(A) = f(f−1 (B)) = Bf(1T̃) = B ∩ 1 = B is IF b~O in (Y, δ). We now assume that A is IF b~C in (T, ψ). Then, 1\f−1 (B) = f−1 (1\B) is IF b~O in (T, ψ). Hence, (1\ B) ∩ f(1T̃) = 1\ B is IF b~O in (Y, δ); thus, 1\(1\ B) = B is IF b~C in (Y, δ). Therefore, (Y, δ) is the IF b~-door space.

### Corollary 4.4

IF b~-open images as well as IF b~-closed images of an IF b~ door space are IF b~ door spaces.

Proof

Because every IF b~O (resp, IF b~C) surjective function is IF b~-quasi-compact, it is an IF b~-door space.

In this study, we introduced the concept of IF b~-door space based on the IF b~ fuzzy set and analyzed its properties. In addition, we introduced the concepts of IF b~-Baire space and IF b~-submaximal space. The properties of the IF b~-door space were presented. The relationship between it and other IF b~-spaces were demonstrated and appropriate decisions were made regarding the results obtained. Furthermore, we proved that an IF b~-door space is a topological property, and that a quasi-compact image of an IF b~-door space is an IF b~-door space. As a generalization, in our next study, we will use other IFSs, such as the IF b- and IF b*-sets, to expand the door space with decision-making regarding the results obtained for each IFS. In addition, we can study other properties of the IF door spaces and the relationships between these spaces and other IFTS.

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

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AbdulGawad A. AL-Qubati received his M.Sc. and Ph.D. degrees in mathematics from the Baghdad University in Iraq, in 1999 and 2004, respectively. He is a professor in the Department of Mathematics at the Hodeidah University in Yemen and is currently a professor in the Department of the Mathematics College of Science and Arts in the Najran University in KSA. His research interests include fuzzy topology, intuitionistic fuzzy topology fuzzy dimensions theory, fuzzy soft topology, intuitionistic and fuzzy functional analysis.

Mohamed El Sayed received his M.Sc. and Ph.D. degrees in mathematics from the Tanta University in Egypt, in 2004 and 2011, respectively. He is currently an associate professor in the Department of Mathematics at the College of Science and Arts, Najran University, KSA. His research interests include general topology and its applications, soft topology, fuzzy topology, and soft rough set.

E-mail: mohammsed@yahoo.com

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 296-302

Published online September 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.3.296

Copyright © The Korean Institute of Intelligent Systems.

## Door Spaces in Intuitionistic Fuzzy Topological Spaces

AbdulGawad. A. Q. Al-Qubati1,2 and Mohamed El Sayed2

1Department of Mathematics, Hodeidah University, Hodeidah, Yemen
2Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia

Correspondence to:A. Q. Al-Qubati (gawad196999@yahoo.com)

Received: September 6, 2021; Revised: April 19, 2022; Accepted: September 7, 2022

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

In this study, a new concept of generalized intuitionistic fuzzy topological space, called intuitionistic fuzzy b~-door space, is introduced and several characterizations of intuitionistic fuzzy b~-door spaces are analyzed. Many examples were introduced to prove the validity of these concepts. Moreover, certain properties and relationships between intuitionistic fuzzy b~-door spaces and other intuitionistic fuzzy topological spaces were investigated.

Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy door space, Intuitionistic fuzzy b~-door space

### 1. Introduction

The notion of an intuitionistic fuzzy set was first defined and further developed by Atanassov [1,2], as a generalization of the fuzzy set proposed by Zadeh [3]. Using the notion of intuitionistic fuzzy sets, Coker [4] introduced intuitionistic fuzzy topological spaces as a generalization of the fuzzy topological spaces proposed by Chang [5]. Several concepts of the fuzzy topological space have been recently extended to intuitionistic fuzzy topological spaces [69]. Parameswari and Thangavelu [10] introduced the concept of b#-open set in ordinary topological spaces and studied several properties of this concept as well as the relationship between the b#-set and other sets. The concept of door spaces was studied in classical topology [11] and then by Donchev [12]. Anjalmose and Thangaraj [13] generalized the concept of door space to fuzzy door spaces.

The aim of this study is to generalize the concept of open and closed sets, called b#-open and b#-closed sets, to intuitionistic fuzzy sets and use them to generalize the concept of topological door spaces to intuitionistic fuzzy topological door spaces. First, we define the concept of intuitionistic fuzzy b~-open and use it to introduce the concept of intuitionistic fuzzy b~-door space. In addition, we introduce the concepts of the intuitionistic fuzzy b~-Baire space and intuitionistic fuzzy b~-submaximal space, as well as their important properties.

Finally, we identify and discuss the relationship between the intuitionistic fuzzy b~-door space and other intuitionistic fuzzy bs-spaces. Furthermore, we prove that the IF b~-door space is a topological property, and that a quasi-compact image of an IF b~-door space is an IF b~-door space.

Section 1 introduces the relative topic, and the basic concepts of the intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces are presented in Section 2. The new concepts for intuitionistic fuzzy b~-door spaces are proposed in Section 3, which is followed by a presentation of the intuitionistic fuzzy functions in b~-door spaces in Section 4. Section 5 presents the conclusion, which briefly summarizes the results and highlights the scope of future research.

### Definition 2.1 ([14])

Let T be a non-empty fixed set. An intuitionistic fuzzy set (IFS) A in T is an object having the form A = {⟨t, μA(t), γA(t)⟩: tT}, where the functions μA:TI and γA : TI denote the degree of membership (μA(t)) and the degree of non-membership (γA(t)) of each element tT to set A, respectively, and 0 ≤ μA(t) + γA(t) ≤ 1, for each tT.

### Definition 2.2 ([14])

Let A and B be IF sets of the form A = {⟨t, μA(t), γA(t)⟩: tT} and B = {⟨t, μB(t), γB(t)⟩: tT}. Then

• (I) AB if μA(t) ≤ μB(t) and νA(t) ≥ νB(t), ∀ tT.

• (II)
• A = B if AB and BA.

(III) Ac = {⟨t, νA(t), μA(t)⟩: tT}.

• (IV) AB = {⟨t, μA(t) ∧ μB(t), νA(t) ∨ νB(t)⟩: tT}.

• (V) AB = {⟨t, μA(t) ∨ μB(t), νA(t) ∧ νB(t)⟩: tT}.

• (VI) 0 = {⟨t, 0, 1⟩: tT}, and 1 = {⟨t, 1, 0⟩: tT}.

• Further details regarding the operations of the IF-sets, IF points, IF functions, and other concepts used in this study can be found in previous references [69,15].

### Definition 2.3 ([4])

An intuitionistic fuzzy topology (IFT) on a non-empty set T is a family Ψ of IF in T that satisfy the following axioms:

• (I) 0, 1ψ.

• (II) If G1, G2ψ, then G1G2ψ.

• (III) If Gψ for each λ ∈ Λ, then ⋃ λ∈ΛGλψ.

• In this case, pair (T, ψ) is called an intuitionistic fuzzy topological space (IFTS) denoted by T, and each intuitionistic fuzzy set in Ψ is an intuitionistic fuzzy open set (IFOS) of T. The complement Ac of IFOS A in IFTS (T, ψ) is the IF closed set (IFCS) in T.

In this study, we introduce the concept of IF b~-open and closed sets as follows:

### Definition 2.4

An IFS A of an IFTS (T, ψ) is an

• (i) intuitionistic fuzzy b~-open set (IF b~OS) if

$A=(cl(int(A)∪(int(cl(A))), and$

• (ii) intuitionistic fuzzy b~-closed set (IF b~CS) if

$A=(cl(int(A))∩(int(cl(A))).$

### Example 2.5

Let T = {a, b} and M be an IF set on T defined as follows:

$M=⟨t, (a/0.5,b/0.5), (a/05,b/0.5)⟩.$

Then the family ψ = {0, 1, M} is an IFTS on T

$IntM=M,Mc=M and cl(int(M))=M.$

Also, cl(M) = M and int(cl(M)) = M.

Therefore, M = (cl(int(M)) ∪ (int(cl(M))), thus M is intuitionistic fuzzy b~-open.

### Example 2.6

Let T = {a, b, c} and M and N be IF sets on T, which are defined as follows:

$M=⟨t, (a/1.,b/0.0,c/0.5), (a/0.0,b/1.0,c/0.5)⟩,N=⟨t, (a/0.0,b/0.0,c/0.5), (a/1.0,b/1.0,c/0.5)⟩.$

Then the family ψ = {0, 1, M, N} is an IFT on T.

$IntM=M, and cl(int(M))=Nc.$

Also, cl(M) = Nc and int(cl(M)) = int(Nc) = M.

Thus, (cl(int(M))∪(int(cl(M))) = Nc and cl(int(M))∩ (int(cl(M))) = M; hence, M is not IF b~-open, but is IF b~-closed.

### Example 2.7

Let T = {a, b} and N be an IF set on T defined as follows:

$N=⟨t, (a/0.4,b/0.3), (a/0.6,b/0.7)⟩.$

Then the family ψ = {0, 1, N} is an IFT on T.

Let K = ⟨t, (a/0.6, b/0.7), (a/0.3, b/0.2)⟩ be an IFS on T.

Clearly, (cl(int(K)) ∩ (int(cl(K))) = ⟨t, (a/0.6b/0.7), (a/0.4, b/0.3)⟩ ⊂ K.

Then, K is not IF b~-closed.

### Definition 2.8

Let A be any IF b~-set in IFTS T. Then, the intuitionistic fuzzy b~-closure (IF b~-closure) and intuitionistic fuzzy b~-interior (IF b~interior) of A are defined as follows:

$IF b∼-cl(A)=∩{F:A⊆F, F is IF b∼CS in T}.IF b∼-int(A)=∪{U:U⊆A, U is IF b∼OS in T}.$

### Definition 2.9 ([7])

Let T be an IFTS and let MT the intuitionistic fuzzy collection.

ψM = {OM : Oτ} is an IFT on T, and pair (M, ψM) is called the IF subspace of an IFTS T.

### Definition 2.10

An IF-point P in an IFTS T is an IF-isolated point if {P} is an IFOS.

### Definition 3.1

An IFTS T is an intuitionistic fuzzy b~-door space (briefly IF b~-door space) if every IF b~-set in T is either IF b~-open or IF b~-closed.

### Example 3.2

Let T = {a, b} and M be an IF set on T, which is defined as follows:

$M=⟨t, (a/0.8,b/0.7), (a/0.2,b/0.3)⟩.$

Then, family ψ = {0, 1, M} is an IFTS on T. The IF set M in T is the IF O set and the complement of an IF setM in T is the IF C set; therefore, the IF sets in T are either IF O or IF C. Therefore, (T, ψ) is an IF door space.

### Example 3.3

Let T = {a, b} and M be an IF set on T defined as follows:

$M=⟨t, (a/0.5,b/0.5), (a/0.5,b/0.5)⟩.$

Then, family ψ = {0, 1, M} is an IFT on T. The IF set M in T is the IF b~O set and the complement of an IF b~-set M in T is the IF b~C-set; therefore, the IF b~-sets in T are either IF b~O or IF b~C. Therefore, (T, ψ) is an IF b~ door space.

### Definition 3.4

Let T be an IFTS, and let A be an IFS in T Then, A is called an IF-nowhere dense set if no non-zero IF-open set B in T exists such that BIF − cl(A). That is,

$IF-int(cl(A))=0T˜.$

### Example 3.5

Let T = {a, b, c} and M, N, and C be IF sets on T, which are defined as follows:

$M=⟨t, (a/0.6, b/0.6, c/0.6), (a/0.3, b/0.3, c/0.4)⟩,N=⟨t, (a/0.5, b/0.5, c/0.5), (a/0.4, b/0.4, c/0.4)⟩,C=⟨t, (a/0.3, b/0.3, c/0.4), (a/0.5, b/0.5, c/0.4)⟩.$

Then, the family ψ = {0, 1, M} is an IFT on T.

$IF-int(cl(Mc))=0T˜, IF-int(cl(C))=0T˜,IF-int(cl(N))=1T˜, and IF-int(cl(M))=1T˜.$

Then, Mc and C are IF-nowhere dense, but M and N are not IF-nowhere dense sets in (T, ψ).

For IF b~-sets, we introduce the definition of IF b~-nowhere dense sets as follows:

### Definition 3.6

Let T be an IFTS, and A be an IFS in T. Then, A is called an IF b~-nowhere dense set if no non-zero IF b~-open set B in T exists such that BIFb-cl(A). That is,

$IFb∼-int(b∼-cl(A))=0T˜.$

### Definition 3.7

Let T be an IFTS. Then, T is called the IF Baire space if IF-int(⋃λ∈ΛAλ) = 0, where A is an IF-nowhere dense set in T.

### Example 3.8

In Example 3.5, Mc and C are IF-nowhere dense sets in (T, ψ). Also, IF − int(McC) = 0. Therefore, (T, ψ) is an IF-Baire space.

For IF b~-sets, we introduce the definition of the IF b~-Baire space as follows:

### Definition 3.9

Let T be an IFTS. Then, T is called the IF b~ Baire space if IFb~-int(⋃λ∈ΛAλ) = 0, where A is an IF b~-nowhere dense set in T.

### Remark 3.10

Example 3.8 will be useful in determining whether the IF sets in Example 3.5 should be replaced the by the IF b~-sets, obtaining the IF b~-nowhere dense set and then the IF b~ Baire space.

### Definition 3.11

Let T be an IFTS. Then, T is an IF b~-submaximal space if every IF b~-dense set is an IF b~-open set in T.

### Definition 3.12

An IF b~S A in an IFTS T is called the IF b~ first category if $(∪i=1∞Ai)=A$, where Ai indicates the IF b~-nowhere dense sets in T. An IF b~S, which is not of the IF b~-first category, is of the IF b~ second category.

### Definition 3.13

An IFTS T is called the IF b~-first-category space $(∪i=1∞Ai)=1T˜$, where Ai indicates the IF b~-nowhere dense sets in T. If T is not an IF b~-first category, then T is said to be an IF b~-second category space.

### Theorem 3.14

Every IF b~-door space T is an IF b~-submax-imal space.

Proof

Let A ⊆ 1 be an IF b~-dense set in T. If A is not IF b~O, then A is an IF b~C because T is an IF b~-door space. Then, A = Ā = 1 and A is an IF b~O (IF b~C). Thus, T is IF b~-submaximal.

The converse of the aforementioned theorem does not need to be true because not all IF b~-open sets are dense, except 1.

### Theorem 3.15

If A is an IF b~-closed set in T with IFb~-int(A) = 0, then A is an IFb~-nowhere dense set in T.

Proof

Let A be an IF b~-closed set in T with IFb~-int(A) = 0 in T. Then, IFb~-cl(A) = A and IFb~-int(A) =; hence IFb~-int(b~-cl(A)) = IFb~-int(A) = 0 in T. Therefore, A is an IFb~-nowhere dense set in T.

The converse of the aforementioned theorem does not need to be true in general; as shown in Example 3.5, C is an IF-nowhere dense set that is not an IF-closed set, as well as for the IF b~-set.

### Theorem 3.16

Every IF subspace of an IF b~ door space (T, ψ) is an IF b~-door space.

Proof

Let (Y, ψY) be a subspace of an IF b~-door space (T, ψ), then YT. Let AY; since (T, ψ) is an IF b~-door space, then A is either IF b~-open or IF b~-closed in (T, ψ), and hence in (Y, ψY). Thus, (Y, ψY) is an IF b~-door space.

### Definition 3.17

An IFTS T is called an IF b~ D-Baire space if every IF b~-first category set in T is an IF b~-nowhere dense set. That is, T is an IF b~ D-Baire space if IFb~-int(cl(A)) = 0 for each IF b~-first category set A in T.

### Proposition 3.18

If T is an IF b~ D-Baire space, then T is an IF b~ Baire space.

Proof

Let A be an IF b~ first-category set in an IF b~ D-Baire space T. Then, $A=(∪i=1∞Ai)=0T˜$, where Ai indicates IF b~-nowhere dense sets and A is an IF b~-nowhere dense set in T. Then, we have IF b~-int(cl(A)) = 0, and IF b~-int(A) ≤ IFb~-intcl(A) implies that IF b~-int(A) = 0. Hence, $IFb∼-int(∪i=1∞Ai)=0T˜$, where Ai indicates IF b~ nowhere dense sets in T. Therefore, T is an IF b~-Baire space.

### Proposition 3.19

Let T be an IF b~-first category space. Then, T is not an IF b~D-Baire space.

Proof

Let T be an IF b~-first-category space. Then, $(∪i=1∞Ai)=1T˜$, where Ai indicates IF b~-nowhere dense sets in T. For the IF b~-first category set 1, we have IFb~-intcl(1) = 0. Hence, T is not an IF b~ D-Baire space.

### Definition 3.20

An IFTS (T, ψ) is an IF b~-Hausdorff if every IF two disjoint points can be separated by IF b~-disjoint open sets.

### Lemma 3.21

Let P = t(α,β) be an IFP in (T, ψ); then, {P} is an IF b~ O only if {P} is an IFOS.

### Lemma 3.22

Let U and V be two IF b~-sets in (T, ψ). If U is an IF b~O and V is an IFO, then UV is an IF b~O in (T, ψ).

### Theorem 3.23

An IF b~-Hausdorff door space T has at most one limit point.

Proof

Let P = t(α, β) and q = y(γ, ρ) be two IF points in T. Because T is an IF b~-Hausdorff, two IF b~-open sets exist, U and V, such that PU, qV, and UV = 0. Because (T, ψ) is an IF b~-door space, set A = (U \ P) ∪ q is either IF b~O or an IF b~C. Then, in the first case by Lemma 3.21, AV = {q} is IF b~-open and by Lemma 3.22 it is IF b~O. In the second case, Ac is IF b~OS; thus, AcU = {P} is IF b~O, and hence IFO. Therefore, in both cases, at least one of the two IF points is an isolated IF point in T, and by contradiction, the request is proved.

### Definition 3.24

An IFTS (T, ψ) is an IF b~-T1/2 space if every IF point {p} is either an IF b~-open set or IF b~ closed set in T.

### Theorem 3.25

Every IFTS (T, ψ) is an IF b~-T1/2 space.

Proof

Let P be an IF point in T. By Propositions 4.3 and 4.4 of [10], {P} is either IF b~-open or IF-nowhere dense because every IF-nowhere dense set is IF b~ CS. Then, {P} is either IF b~ OS or IF b~ CS. Thus, T is an IF b~-T1/2 space.

### Theorem 3.26

Every IF door space (T, ψ) is an IF b~-space.

Proof

Clearly.

The converse of the aforementioned theorem does not need to be true in general because not all IF bs-OS or IF bs-CS are IFO or IFC, respectively; see Example 2.6.

### Theorem 3.27

Let (T, ψ) be an IFTS; then, the following are equivalents.

• (i) (T, ψ) is an IF-discrete space.

• (ii) (T, ψ) is an IF b~-door space.

Proof

(i) ⇒ (ii) Suppose that (T, ψ) is an IF discrete; then, every IF set is an IFO or IFC, and then IF b~-open or IF b~-closed. Thus, (T, ψ) is an IF b~-door space.

(ii) ⇒ (i) Let P = x(α,β) be an IFP in T. Then, if {P} is not IF b~-open, then {P} is IF b~-closed and {P} = (IFb~-cl(int(P)) ∩ (IFb~-int(cl(P))). Because IFb~int({P}) is not empty and {P} = (IFb~-cl(P)), then {P} = (IFb~-cl(int(P))∩(IFb~-int(p)) = IFb~-int({P}). In both cases, {P} is an IFO. Therefore, (T, ψ) is an IF discrete space.

### Definition 3.28

An IF b~-TS (T, ψ) is irreducible if every IF b~ set in (T, ψ) is IF b~ connected, equivalent to every non-void IF b~ OS in (T, ψ) being dense.

The following theorem introduces an additional condition for the converse of Theorem 3.14 to be true.

### Theorem 3.29

Every IF irreducible submaximal space (T, ψ) is an IF b~-door space.

Proof

Let A be an IF b~-set in (T, ψ). If A is IF b~-dense, then because (T, ψ) is IF b~-submaximal, A is IF b~-open; if A is not IF b~-dense, then we can find a non-void IF b~O set BAc. Because (T, ψ) is irreducible, B and Ac are IF b~-dense. In addition, because (T, ψ) is IF b~-submaximal, Ac is IF b~ O or equivalently, A is IF b~ C. Thus, in any case, A is either IF b~ O or IF b~ C. Therefore, (T, ψ) is an IF b~-door space.

### Definition 3.30

Let (T, ψ) be an IFTS. An intuitionistic fuzzy family I in T is called an intuitionistic fuzzy ideal (IFI) if it is hereditary and has additivity that satisfies the following properties:

• (i) A ∈ IFI and BA implies B ∈ IFIA (hereditary),

• (ii) A ∈ IFI and B ∈ IFI, then AB ∈ IFI (finite additivity).

Let us denote Icd as the IFI of the intuitionistic fuzzy closed discrete sets in T.

### Definition 3.31

An IFTS topological space (T, ψ, I) is called an IFI b~-door space if every IF-set in T is either IF b~-open, IF b~-closed, or belongs to IFI.

### Remark 3.32

Clearly, every intuitionistic fuzzy door space is an intuitionistic fuzzy IFI b~-door space.

### Definition 3.33

An IFTS T is called IF b~-extremally disconnected if the IF b~-closure of an IF b~-open set is IF b~-open.

### Proposition 3.34

Let T be an IFTS. Then, the following conditions are equivalent:

• (i) T is an IF b~-extremally disconnected b~-door space.

• (ii) Every IFS A in T is either IF b~-open, or both IF b~-closed and IF discrete.

Proof

(i) ⇒ (ii) First, IFS A belongs to Icd only if A has no IF limit point. Suppose that A is not IF b~ OS and does not belong to d(A) ≠ ∅. Because T is an IF b~-door space, A is IF b~-closed. Let td(A) ⊆ A, and S = Ac ∪ {p}. Because P is an IF limit point of A, S cannot be IF b~-open; for in that case, S would be an IF b~-open neighborhood of P containing a point of A different than P. Clearly, this is impossible because SA = {P}. Because S is not IF b~-open and because T is an IF b~-door space, S is IF b~-closed. Thus, Ac ∪ {p} = IF-cl(Ac) ∪ IF-cl{p}; hence, AcIF-cl(Ac) ⊆ Ac ∪ {p}. By assumption, because T is IF b~-extremally disconnected, IF-cl(Ac) is IF b~-open. Because S is IF b~-closed, AcIF-cl(Ac) ⊆ (Ac) ∪ {p} = S. Thus, Ac = IF-cl(Ac) or equivalently, A is IF b~-open. Thus, this contradicts with the assumption that A is not IF b~-open.

(ii) ⇒ (i) Let T be an IF b~-door space. We must demonstrate that T is IF b~-extremally disconnected. Let AT be IF b~-open. Then, IF-cl(A) = Ad(A)). If d(A) = ∅, then I-cl(A) = A is IF b~-open. If d(A) ≠ ∅, then d(IF-cl(A)) ≠ ∅ and IF-cl(A) are IF b~-open, which follows (ii).

### Definition 4.1

Let f : (T, ψ) → (Y, δ) be an IF function. Then, f is IF b~-quasi compact if A ⊆ 1 is IF b~ O in (T, ψ) such that f−1(f(A)) = A. Then, f(A) is IF b~ O in (Y, δ).

### Theorem 4.2

An IF b~-door space has a topological property.

Proof

Let f : (T, ψ) → (Y, δ) be an IF b~-homemorphism from an IF b~-door space (T, ψ) into another IF b~-TS (Y, δ). Let A be IF b~S in (Y, δ), that is, AY. Then, f−1(A) is IF b~S in (T, ψ), that is, f−1(A) ⊆ T. Because (T, ψ) is an IF b~-door space, f−1(A) is IF b~O or an IF b~CS in (T, ψ), because f(f−1(A)) = A. Then, A is either IF b~OS or IF b~CS in (Y, δ). Thus, (Y, δ) is the IF b~-door space.

### Theorem 4.3

An IF-quasi-compact image of an IF b~-door space is an IF b~-door space.

Proof

Let f : (T, ψ) → (Y, δ) be an IF-quasi compact function from an IF b~-door space (T, ψ) into another IF b~-TS (Y, δ). Let B be IF b~S in (Y, δ). We must prove that B is either IF b~O or IF b~C in (Y, δ). Because (T, ψ) is an IF b~-door space and f−1(B) = A, then A is either IF b~O or IF b~C in (T, ψ). Suppose that A is IF b~O, and clearly f(A) ⊆ B. Thus, f−1f(A) ⊆ f−1(B) = Af−1f(A) or equivalently, A = f−1f(A). By assumption, f(A) = f(f−1 (B)) = Bf(1T̃) = B ∩ 1 = B is IF b~O in (Y, δ). We now assume that A is IF b~C in (T, ψ). Then, 1\f−1 (B) = f−1 (1\B) is IF b~O in (T, ψ). Hence, (1\ B) ∩ f(1T̃) = 1\ B is IF b~O in (Y, δ); thus, 1\(1\ B) = B is IF b~C in (Y, δ). Therefore, (Y, δ) is the IF b~-door space.

### Corollary 4.4

IF b~-open images as well as IF b~-closed images of an IF b~ door space are IF b~ door spaces.

Proof

Because every IF b~O (resp, IF b~C) surjective function is IF b~-quasi-compact, it is an IF b~-door space.

### 5. Conclusion

In this study, we introduced the concept of IF b~-door space based on the IF b~ fuzzy set and analyzed its properties. In addition, we introduced the concepts of IF b~-Baire space and IF b~-submaximal space. The properties of the IF b~-door space were presented. The relationship between it and other IF b~-spaces were demonstrated and appropriate decisions were made regarding the results obtained. Furthermore, we proved that an IF b~-door space is a topological property, and that a quasi-compact image of an IF b~-door space is an IF b~-door space. As a generalization, in our next study, we will use other IFSs, such as the IF b- and IF b*-sets, to expand the door space with decision-making regarding the results obtained for each IFS. In addition, we can study other properties of the IF door spaces and the relationships between these spaces and other IFTS.

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