CONNECTEDNESS IN JÄGER - ŠOSTAK’S I-FUZZY TOPOLOGICAL SPACES

G.Jäger [Compactness and connectedness as absolute properties in fuzzy topological spaces, Fuzzy sets and Systems 94(1998) 405—401] introduced a kind of (general) fuzzy topological space. In this paper, we propose a new kind of topological space in Šostak’s sense, called JägerŠostak’s I-fuzzy topological space, which reduced to Jäger’s (general) fuzzy topological to two-valued logic. After that for each fuzzy subset of Jäger-Šostak’s I-fuzzy topological space, we define a degree of connectedness, which overcome the deficit of study for the whole space a degree of being connected in public papers, and establish two characteristic theorems of the degree of being connectedness. Doing so we find that the degree of connectedness is an absolute property in JägerŠostak’s I-fuzzy topology.


Introduction and preliminaries
Since C.L. Chang [1] introduced fuzzy theory into topology, many authors discussed various aspects of fuzzy ( I-fuzzy, fuzzifying) topology.For connectedness of fuzzy topological space, many authors devoted their researches to the topics, such as [4,6,11].In [6], the author proposed a concept of connectedness and showed it is an absolute property in (general) fuzzy topology space.Now we'll study degree of connectedness of Jäger-Šostak's I-fuzzy topological space in more general form.
The contents are arranged as follows.In next section, we introduce basic concepts, such as Jäger-Šostak's I-fuzzy topology space, morphisms and its degree of continuity.Moreover, we give the definition of Jäger-Šostak's I-fuzzy subspace, which is the crucial part of this paper so that the degree of connectedness defined in the paper is an absolute property.In last section, for each fuzzy subset of Jäger-Šostak's I-fuzzy topological space, we introduce a degree of connectedness and show its characteristic theorem of the degree of being connectedness.After that, we give another characteristic theorem of the degree of connectedness by the degree of separation.Finally, we conclude that the degree of connectedness defined in the paper, is an absolute property in Jäger-Šostak's I-fuzzy topology.
In this paper, X, Y, • • • are nonempty sets.A fuzzy subset on a universal set X is a mapping from X to the real unit interval [0,1].All fuzzy subsets on X is denoted by I X and for a subset Y ⊆ X, the characteristic function of Y is denoted by 1 Y .Fuzzy subsets are denoted by capital letters A, B, C, • • • ∈ [0, 1] X .And the greatest element of I X is denoted by 1 X and the least element of I X is denoted by 0 X .Note that we don't distinguish a number λ ∈ [0, 1] and the constant function λ : X → [0, 1] such that λ(x) = λ for all x ∈ X.It's assumed that the reader is acquired with the usual definitions and notations in fuzzy set theory and fuzzy topology.Given a fuzzy subset A ∈ [0, 1] X , we denote For the complement of B relative A, we have following proposition to describe its basic properties and omit their proofs.Proposition 1.1.For any fuzzy subset A ∈ I X , the following hold: (1) For each

Jäger-Šostak's I-fuzzy topological spaces
Jäger [6] proposed a concept of a fuzzy topology on a fuzzy subsets A so that a lot of topological questions can be researched on any fuzzy subsets in place of a whole space.According to the ideas of L-fuzzy topology in the sense of [5], we generalize Jäger's fuzzy topology to a fuzzy subset as follows.
is called a Jäger-Šostak's I-fuzzy topology on a fuzzy subset A iff τ satisfies the following conditions: The pair (A, τ ) is called a Jäger-Šostak's I-fuzzy topological space (JSIfts, in short).The value of τ (B) for every B ∈ F X (A) is interpreted as the degree to which B is open relative to τ , the value of τ (h A H) for each H ∈ F X (A) is the degree to which H is closed relative to τ . 2 Remarks 2.2.Assume that τ : F X (A) → I is a Jäger-Šostak's I-fuzzy topology.Then a (general) fuzzy topology T τ on the fuzzy subset A in the sense of [6] can be induced by τ , that is, T τ ={U : τ (U ) = 1} satisfies the following conditions: Conversely, if T ⊆ I X is a general fuzzy topology, then a Jäger-Šostak's I-fuzzy topology τ T is obtained by defining τ T (U) = 1 whenever U ∈ T and τ T (U ) = 0 otherwise.Therefore, Jäger-Šostak's I-fuzzy topology is more general than a (general) fuzzy topology.2 In order to obtain morphisms between two JSI-fts's, we have to restrict function f : X→Y to fuzzy subsets as Jäger [6] did.Given such a function f : X→Y and A ∈ [0, 1] X , we called the mapping g  (Jäger [6]).We say that g : A → B is a fuzzy mapping from A to B if and only if there exists a function f : Now we propose the definition of morphism between two JSI-fts's.
Definition 2.5.In the situation above, for each fuzzy mapping g : (A, τ ) → (B, δ), the degree to which g is continuous, is defined ´ ¶¾ . 2 Thus, a fuzzy mapping g : A → B is continuous if and only if its degree to which g is continuous is 1.
Proposition 2.6.The follow conclusions hold: Proof.(i) is obvious.In order to show (ii), we show inequality in two cases: ´.
It follows that ´ ¶¾ ´ ¶¾ For the proofs of (iii), note that for each In the following, we are going to introduce a suitable form of subspace of a Jäger-Šostak's I-fuzzy topological space. Let the Jäger-Šostak's I-fuzzy subspace topology on B induced by τ , and call the pair (B, τ c B ) the Jäger-Šostak's I-fuzzy subspace of (A, τ ).When τ is a fuzzy topology on A in the sense of G. Jäger [6], the definition is identified with the definition of fuzzy subspace of G. Jäger [6].Actually, this subspace concept was first used in Liu and Luo [7] but was so far not reached general interest in I-fuzzy setting.
Proposition 2.7.For any JSI-fts (A, τ ), we have ( C (H) is true, and hence the conclusion is deduced from the arbitrariness of H.
First of all, for each H ∈ F X (C) we have that On the other hand, when letting Therefore, it follows that for each Proposition 2.8.For any (A, τ ) JSI-fts and any fuzzy subset B ∈ I X with B ≤ A, it holds Proof.This is obvious by the definition of According to Remarks 2.2, when a T is a (general) fuzzy topology on a fuzzy subset A and we identify the Jäger-Šostak's I-fuzzy topology τ T induced by T with T , then the results stated in [6] can be described as follows.
Corollary 2.9.Let (A, T ) be a (general) fuzzy topological space and Corollary 2.10.Let (A, T ) be a (general) fuzzy topological space and B ≤ A. We then have for each M ≤ B, T c B (h B M ) = 1 if and only if there exists a H ≤ A such that T (h A H) = 1 and M = B ∧ H. 2 For an I-fuzzy topology S on a set X in the sense of Höhle and Šostak [5], there exists a good way defining I-fuzzy subspace topology on a crisp subset Y ⊆ X [2,10].In the following, we hope to show this kind of I-fuzzy subspace topology could be described as a special case of Jäger-Šostak's Ifuzzy sub-topology of the paper.Before we state the results we quote the definition below.
Definition 2.11 (Höhle and Šostack [5]).An I-fuzzy topology on a set X is a map S : I X → I such that (I-FO1) S(λ) = 1 for every constant fuzzy subset λ : X → I; If S is an I-fuzzy topology on X, then we say that (I X , S) is an I-fuzzy topological space (I-fts, in short). 2 Obviously, every I-fuzzy topological space (I X , S) can be identified as a Jäger-Šostak's I-fuzzy topological space (1 X , S) if necessary.
In [7] and later [2], a concept of I-fuzzy sub-topology was proposed, in details, for a crisp subset Y ⊆ X, an I-fuzzy sub-topology S Y : I Y → I on Y induced by S is defined as follows: for each and (I Y , S Y ) is called the I-fuzzy topological subspace of (I X , S). Proposition 2.12.Let (I X , S) be an I-fts and Proof.Note that we identified F X (B) with F Y (B).The equality S c B = S Y is obtained by the following formulas,∀H ∈ F Y (B), Remark 2.13.Proposition 3.4 shows that for a crisp subset B = 1 Y of an I-fts (I X , S), the subspace topologies S c B and S Y [2, 10] coincide.2

The degree of I-fuzzy connectedness of a fuzzy set
In classical topology, there are many ways to describe the definition of connectedness.For example, no existing separation, each continuous function into the discrete two points space being constant, K. Fan theorem's way, etc.In [6], G. Jäger gave a definition of connectedness of a fuzzy subset in a (general) fuzzy topological space similar to the one in classical topology, i.e., a topological space is connected iff each continuous function into discrete two points space is constant, and it was proved by Jäger, that this kind of connectedness of a fuzzy subset is an absolute property.The definition adopted in [6] was proved to be equivalent to Pu and Liu's definition [9] even for fuzzy subsets.Jäger's manner defining connectedness of a fuzzy subset also can be found in Lowen and Srivastava [8] in the case that the fuzzy subset is the whole space.
The degree of connectedness of fuzzy subsets presented here in Jäger-Šostak's I-fuzzy topological spaces will be introduced in this section.If the degree of connectedness of a fuzzy subset is the designed truth value 1, we then go back to Jäger's definition [6] in two-valued logic.We will prove here our general I-fuzzy degree of connectedness is absolute.
In the following, we call H t (A) := , where 1 2 is the characteristic function of 2. The discrete Jäger-Šostak's I-fuzzy topology on 2 α is denoted by T α , which means that T α (B) = 1 for all B ≤ 2 α .Furthermore, for A ∈ I X , B ∈ I Y , we call a fuzzy mapping The set of all fuzzy mappings from A to B which isn't constant, is denoted by Sur(A 0 , Y ).P → I, called a general I-fuzzy degree of connectedness, is given as follows: ´, where α = H t (A). 2 Corollary 3.2.Let (A, τ ) be JSI-fts and g be a fuzzy mapping from A to 2 Ht(A) .Then c(g) = Proof.By the definition above, Let us give the first characteristic theorem of the general I-fuzzy degree of connectedness as follows.
Proof.According to the definition of con G (A, τ ), we need to prove ´.
To complete the target here, let us assume that ín the present time.Thus our target change to prove α = β.The strategy here from is to show both α ≤ β and β ≤ α.Let γ be any real number such that 0 ≤ γ < β.According to the definition of β there is a non-void proper subset E of A 0 such that ) is continuous and surjective.Then for t : 0 ≤ t ≤ H t (A), we have ´ ¶ ´ ¶! > γ for each s with 0 ≤ s ≤ H t (A).Everyone know that there are t D , s C with the property of 0 By the arbitrariness of γ, we get β ≤ α.
Conversely, let γ be any real number with 0 ≤ γ < α now.It follows from the definition of α that there exists a mapping g ∈ Sur(A 0 , 2), and here g = f | A such that there exist two non-void disjoint subsets E 1 and Therefore, α ≤ β holds from the arbitrariness of γ. 2 Corollary 3.4.For a JSI-fts (A, τ ), Proof.This is obvious by Theorem ) is a JSI-fts, i.e., τ is an I-fuzzy topology on X (see Definition 2.11).Then it holds Con(I X , τ) = Con G (1 X , τ), where defined by Fang and Yue [3] and its Proof.Note that for each pair (B, C) ∈ I X × I X , it holds that (B, C) ∈ D if and only if B = 1 B 0 , C = 1 C 0 , B 0 ∪ C 0 = X and B 0 ∩ C 0 = ∅.Thus using Corollary 3.4, we have Remark 3.6.By Corollary 3.5, we have generalized I-fuzzy connection proposed in [3] to more general form, that is , the general I-fuzzy degree of connectedness for any fuzzy subsets. 2 As pointed out at the beginning of the section, there are many ways to describe the definition of connectedness.Now we try to characterize the general I-fuzzy degree of connectedness by the ideas of separation.Now, a definition of separation in fuzzy setting is needed.Definition 3.7.Let (A, τ ) be a JSI-fts.A mapping S G : → I, called a general I-fuzzy degree of separation, is given as follows: Following the definition above, the characteristic theorem of general Ifuzzy degree of connectedness using the general I-fuzzy degree of separation is obtained.Theorem 3.8.Let (A, τ ) be a JSI-fts, D ∈ F X (A).Then Especially when D = A, it holds Proof.By Theorem 3.3 , It suffices to show that Now letting γ be any positive real number with the property of then there exists a non-void proper subset E of D 0 such that It follows from the arbitrariness of γ, that Conversely, let γ be any positive real number so that Then it follows that there exist M, N ∈ F Therefore, Finally, we complete the proofs by the two inequalities (3.1) and (3.2) above. 2 At the end of the section, we want to study the absolute property of the general I-fuzzy degree of connectedness defined in the paper.Recall that we call a property P is absolute in classical topology if for all subspaces Z ⊆ Y ⊆ X of a topological space (X, T ), Z has the property P as a subspace of Y if and only if Z has the property P as a subspace of X.In I-fuzzy setting we propose a definition as follows: a degree mapping Γ describing a property P is said to be absolute if and only if for all subspaces C ≤ B ≤ A of a Jäger-Šostak's I-fuzzy topological space (A, τ ), the true value Γ(C) having property P as a subspace of B is equal to the true value Γ(C) having property P as a subspace of A. In this way, we have Theorem 3.9.The general I-fuzzy degree of connectedness is absolute.), that is, the degree mapping Con G stating general I-fuzzy degree of connectedness is absolute, as desired.

Conclusions.
In this paper, we propose a new concept of Jäger-Šostak's I-fuzzy topological space for any fuzzy subset in place of a whole space.Obviously, this is a new kind of fuzzy topology, namely Jäger-Šostak's I-fuzzy topology with the fixed basis L = I, the real unit interval.Some of basic concepts are established, such as subspaces and suitable morphisms.In the framework of Jäger-Šostak's I-fuzzy topology, we research the general I-fuzzy degree of connectedness of a fuzzy subset in Jäger-Šostak's I-fuzzy topology, and its characteristic theorems are obtained.Moreover, we show that the general I-fuzzy degree of connectedness defined in the paper is absolute, which reflect the concept of subspace in the paper is rational.We think there are a lot of questions, such as Neighborhood systems, Moore Smith's convergence, separation axioms, compactness, etc., needed to be settled.We will exploit them in our following works.
the restriction of f on A, where for all y ∈ Y , f (B)(y) = sup{B(x) : f (x) = y} as defined by Zadeh.The inverse imagine of a fuzzy subset D

Definition 3 . 1 .
Let P denote the class of all Jäger-Šostak's I-fuzzy topological spaces.A mapping Con G : Proof.Let (A, τ ) be a JSI-fts and C ≤ B ≤ A. Thus there are two topologies (τ c B ) c C and τ c C on C. By proposition 2.7, we know that (τ c B ) c C = τ c C .It follows from the definition of the general I-fuzzy degree of connectedness that Con G (C, τ c C ) = Con G (C, (τ c B ) c C