Double-Connected Intuitionistic Space in Double Intuitionistic Topological Spaces

Te structures of the families of fuzzy sets that arise out of various notions of openness and closeness in a double fuzzy topological space are explored. Tis research is to present a new portion of space (Double-connected intuitionistic space) in Double intuitionistic topological spaces. Trough these concepts, we advance some of their characteristics and relate to themselves.


Introduction
Ajmal and Kohli [1] explain connectedness in fuzzy topological spaces. Yasry [2] investigated lectures in advanced topology. El-Hamed et al. [3] presented Double connected spaces. Kandil et al. [4] investigated on four (intuitionistic) topological spaces. Kendal et al. [5] studied on four (intuitionistic) compact space. Te concept was used to defne intuitionistic sets and the intuitionistic gradation of openness by Coker [6][7][8]. Atanassov and Stoeva [9] describe Intuitionistic fuzzy sets. Atanassov [10] studied more intuitionistic fuzzy sets. Ozcelik and Narli [11] introduced the concept of submaximal intuitionistic topological spaces. Tantawy et al. [12] researched soft connections with double spaces. Selma and Coker [13] examined the concept of connectedness in intuitionistic fuzzy special topological spaces. Levine [14,15] introduced generalized closed sets in topology. Te concept of a fuzzy set was presented by Zadeh in his classic paper from 1965 and 2002 [14,16]. After that, we introduce a new class of sets in DITS, namely the Double connected set, the separated Double I sets, the strongly Double connected, Double CO connected I space, and the Double I component. We also presented several examples of each type and concluded that there are relationships between them that were presented through theorems.
(3) ζ c � 〈x.ζ 2 .ζ 1 〉. (2) i 〉 [7]. Let x be a nonempty set, an intuitionistic set B (I S, for short) is an object having g the form B � 〈x.B 1 .B 2 〉, where B 1 and B 2 are disjoint subset of x. Te set B 1 is called set of members of B, while B 2 is called set of nonmembers of B [7]. An intuitionistic topology (IT, for short) on a nonempty set x is a family T of IS in X containing ∅ .x and closed under arbitrary unions and fnitely intersections. Te pair (x.T) is called ITS [11]. Let x ≠ ∅.
(1) A Double-set (D-set, for short) U is an ordered pair [4]. Let x be a nonempty set. Te family η of D-sets in x is called a double topology on x if it satisfes the following axioms: Te complement of open D-set is called closed D-set [4]. Let x be a nonempty set: is a DTS, which is called discrete DTS [4]. Let (X, η) be a DTS and z ∈ D (X). Te double closure of U, denoted by cl η (U) defned by cl η (U) � ∩ {ϑ: ϑ ∈ η c and U ⊆ ϑ} [2,4]. Let X nonempty set, w ∈ X a fxed element in X, and let M � 〈x.M 1 .M 2 〉 be an intuitionistic set (IS, for short).
. A topological space X is connected if it cannot be written as X � X 1 ∪ X 2 , where X 1 and X 2 are both open and X 1 ∩ X 2 � ∅, otherwise X is called disconnected [2]. Let (X, η) be a DTS and Ƴ be a nonempty subset of X. Ten, η Ƴ � {q ∩ Ƴ: q ∈ M and Ƴ � (Ƴ, Ƴ)} is a double topology on Ƴ. Te DTS (Ƴ, η Ƴ ) is called a double topological subspace of (X, η) (DT-subspace, for short) [4]. Let (X, η) be a DTS and let ɣ, ɧ∈D (X): ɣ, ɧ are said to be separated double sets (separated Dsets, for short) if cl η (ɣ) ∩ ɧ � ∅ and cl η (ɧ) ∩ ɣ � ∅ [3]. Let (X, η) be a DTS, and let be a nonempty subset of X. If there exist two nonempty separated D-sets ɣ, ɧ∈D(X) such that ɣ ∪ ɧ � Ɲ, then the D-sets ɣ and ɧ form a Dseparation of Ɲ and it is said to be double disconnected set (D-disconnected set, for short). Otherwise, Ɲ is said to be double connected set (d-connected set, for short) [3]. Te DTS (X, Π) is said to be (1) C 5 -disconnected, if (X, Π) has a proper open and closed D-set in Π.

Double-Connected Intuitionistic Space in DITS
In this section, we defne new kinds of x is called Double connected I space, separated Double I sets, strongly Double connected, Double CO connected I space, and Double I component in Double intuitionistic topological spaces, and joined to other kinds of sets that are defned in this work. We start this section by the following defnitions: called the universal Double I set, and the Double I set Each element of Ѡ is called a DIOS in x. Te complement of DIOS is called DICS. Now, we want to introduce the important theorem to construct the Double intuitionistic topological spaces.
Te following example shows that the converse is not true: is not separated Double I sets.
Te following defnition of Double-connected intuitionistic sets in DITS: □ Defnition 4. Let (x, Ѡ) be a DITS; let (k, Z) be a nonempty subset of X. If there exist two nonempty separated Double I sets (L, Ӻ), (I, Ω) ∈ DI (X) such that (L, Ӻ) ∪ (I, Ω) � (k, Z), then the Double I sets (L, Ӻ) and (I, Ω) form a Double separation of (k, Z) and it is said to be Double disconnected intuitionistic sets (Double disconnected I sets, for short) Otherwise, (k, Z) is said to be Double-connected intuitionistic sets (Double connected I sets, for short).