WEAKLY INDUCED MODIFICATIONS OF I-FUZZY TOPOLOGIES

Since Chang [2] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. It is well known that weakly induced and induced topological spaces play an important role in L-topological spaces (see book [8]). According to their value ranges, L-topological spaces form different categories. Clearly, the investigation on their relationships is certainly important and necessary. Lowen was the first author to study the relations between I-topological spaces and classical topological spaces. He introduced two well-known functors—ω and ι. Later, these functors, named Lowen functors, were extended by different authors [7, 12] for various kinds of lattices studying the relations between L-TOP and TOP. However, in a completely different direction, Höhle [4] created the notion of a topology being viewed as an L-subset of a powerset. Then Kubiak [6] and Šostak [11] independently extended Höhle’s notion to L-subsets of LX . From a logical point of view, Ying [13] introduced fuzzifying topological spaces (Ying’s fuzzifying topology is similar to Höhle’s topology). In order to discuss the relations between fuzzifying topologies and I-fuzzy topologies, the authors studied Lowen functors in I-fuzzy topological spaces in a KubiakŠostak sense and introduced induced I-fuzzy topological spaces in [15]. Zhang and Liu [17] studied weakly induced modifications of L-topologies. The aim of this paper is to study weakly induced I-fuzzy topological spaces and the weakly induced modifications of I-fuzzy topologies. This paper is organized as follows. In Section 1, we give some preliminary concepts and properties. Two kinds of weakly induced modifications are introduced in Section 2.


Introduction and preliminaries
Since Chang [2] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology.It is well known that weakly induced and induced topological spaces play an important role in L-topological spaces (see book [8]).According to their value ranges, L-topological spaces form different categories.Clearly, the investigation on their relationships is certainly important and necessary.Lowen was the first author to study the relations between I-topological spaces and classical topological spaces.He introduced two well-known functors-ω and ι.Later, these functors, named Lowen functors, were extended by different authors [7,12] for various kinds of lattices studying the relations between L-TOP and TOP.
However, in a completely different direction, Höhle [4] created the notion of a topology being viewed as an L-subset of a powerset.Then Kubiak [6] and Šostak [11] independently extended Höhle's notion to L-subsets of L X .From a logical point of view, Ying [13] introduced fuzzifying topological spaces (Ying's fuzzifying topology is similar to Höhle's topology).In order to discuss the relations between fuzzifying topologies and I-fuzzy topologies, the authors studied Lowen functors in I-fuzzy topological spaces in a Kubiak-Šostak sense and introduced induced I-fuzzy topological spaces in [15].Zhang and Liu [17] studied weakly induced modifications of L-topologies.The aim of this paper is to study weakly induced I-fuzzy topological spaces and the weakly induced modifications of I-fuzzy topologies.
This paper is organized as follows.In Section 1, we give some preliminary concepts and properties.Two kinds of weakly induced modifications are introduced in Section 2.
2 Weakly induced modifications of I-fuzzy topologies We prove that I-WIFTOP-the category of weakly induced I-fuzzy topological spacesis a reflective and coreflective full subcategory of I-FTOP.Finally, in Section 3, we discuss the relationship between several important categories.
In this paper, X is a nonempty set and I = [0,1], I 0 = [0,1).The family of all fuzzy sets on X will be denoted by I X .By 0 X and 1 X , we denote, respectively, the constant fuzzy set on X taking the values 0 and 1.Let σ r (A) = {x | A(x) > r} for r ∈ I and A ∈ I X .U ∈ P(X), 1 U denotes the characteristic function of U, that is, 1 U (x) = 1 when x ∈ U and 1 U (x) = 0 when x ∈ U.For the notions about categories, please refer to [1,5,9].Definition 1.1 [4,13].A fuzzifying topology on X is a map ξ : P(X) → I satisfying the following axioms: If ξ is a fuzzifying topology on X, the pair (X,ξ) is called a fuzzifying topological space.A fuzzifying continuous map between fuzzifying topological spaces (X,ξ) and (Y ,η) is a map f : X → Y such that ξ( f ← (U)) ≥ η(U) for all U ∈ P(Y ).The category of fuzzifying topological spaces and fuzzifying continuous maps is denoted by FYS.Let FYS(X) denote the set of all fuzzifying topologies on X. Definition 1.2 [6,11].An I-fuzzy topology on a set X is defined to be a map -: Ifis an I-fuzzy topology on X, the pair (I X ,-) is called an I-fuzzy topological space.An I-fuzzy continuous map between I-fuzzy topological spaces (I X ,-) and (I Y ,) is a map f : ) (following the notation in [10]).The category of I-fuzzy topological spaces and I-fuzzy continuous maps is denoted by I-FTOP.Let I-FTOP(X) denote the set of all I-fuzzy topologies on X. Definition 1.3 [13].Let ξ be a fuzzifying topology on X, Ꮾ : P(X) → I, and Ꮾ ≤ -.Ꮾ is called a base of ξ if Ꮾ satisfies the following condition: where the expression λ∈∧ Vλ=U λ∈∧ Ꮾ(V λ ) will be denoted by Ꮾ ( ) (U), that is, ξ = Ꮾ ( ) .
A map φ : P(X) → I is called a subbase of ξ if φ ( ) : P(X) → I defined by φ ( ) (U) = ( )λ∈J Vλ=U λ∈J φ(V λ ) for all U ∈ P(X) is a base, where ( ) stands for "finite intersection."φ : P(X) → I is a subbase of one fuzzifying topology if and only if φ ( ) (X) = 1.Definition 1.4 [14].Let {(X t ,ξ t )} t∈T be a family of fuzzifying topological spaces and let P t : t∈T X t → X t be the projection.Then the fuzzifying topology whose subbase is Y. Yue and J. Fang 3 defined by is called the product topology of {ξ t | t ∈ T}, denoted by t∈T ξ t .( t∈T X t , t∈T ξ t ) is called the product space of {(X t ,ξ t )} t∈T .
Fang and Yue [3] extended the above definitions and results to I-fuzzy topological spaces.For more explicitly, we list them as follows.
(3) Let {(I Xt ,t )} t∈T be a family of I-fuzzy topological spaces and let P t : t∈T X t → X t be the projection.Then the I-fuzzy topology whose subbase is defined by is called the product topology of {-t | t ∈ T}, denoted by t∈Tt .(I t∈T Xt , t∈Tt ) is called the product space of {(I Xt ,t )} t∈T .
Definition 1.5.Let {(I Xt ,t )} t∈T be a family of I-fuzzy topological spaces, let different X t s be disjoint and X = t∈T X t , and let -: I X → I be defined as follows: Then it is easy to verify thatis an I-fuzzy topology on X, andis called the sum topology of {-t } t∈T , denoted by t∈Tt .
Definition 1.6.Let (I X ,-) be an I-fuzzy topological space and let f : X → Y be a surjective map.Define the I-fuzzy quotient topology -/ f → I ofwith respect to f by It is easy to verify that -/ f → I is an I-fuzzy topology on Y .(I Y ,-/ f → I ) is called the I-fuzzy quotient space of (I X ,-) with respect to f and f → I is called an I-fuzzy quotient map.Definition 1.7 [9].Let (I X ,-) be an I-fuzzy topological space and Lemma 1.8 [5].I-FTOP(X) is a complete lattice.
Using the similar argument in [5], it is easy to show that FYS(X) is also a complete lattice.

Weakly induced modifications of I-fuzzy topologies
The purpose of this section is to study weakly induced I-fuzzy topological spaces and the weakly induced modifications of I-fuzzy topologies.
) for all A ∈ I X , then (I X ,-) is called a weakly induced I-fuzzy topological space.Let I-WIFTOP denote the category of weakly induced I-fuzzy topological spaces.
Example 2.2.Let ξ be a fuzzifying topology on X. Defineξ : I X → I as follows: It is easy to check thatξ is an I-fuzzy topology on X and it is weakly induced.Specially, is weakly induced, where Thenis a weakly induced I-fuzzy topology on X.
In the following discussion, we will give the right adjoint functor and left adjoint functor of the inclusion functor i : I-WIFTOP → I-FTOP, and show that I-WIFTOP is a reflective and coreflective full subcategory of I-FTOP.Lemma 2.4.Let (I X ,-) be an I-fuzzy topological space and let - * : I X → I be defined by (
Y. Yue and J. Fang 5 Proof.It is routine to prove that - * is an I-fuzzy topology on X.The following computation can show that - * is weakly induced: (2.4) Let be any weakly induced I-fuzzy topology on X satisfying ≤ -.We need to prove that ≤ - * .Since is weakly induced, we have (A) ≤ r∈I0 (1 σr (A) ) for all A ∈ I X .Hence we get that thus the conclusion.
Lemma 2.5.Let (I Y ,-) be weakly induced and let (I X ,) be an I-fuzzy topological space. Then Proof.The sufficiency is obvious and it needs to show the necessity.Let (2.6) Therefore, f → I : (I X , * ) → (I Y ,-) is I-fuzzy continuous.Remark 2.6.From Lemma 2.5, we also can get that Hence we know that (•) * is a functor from I-FTOP to I-WIFTOP.Furthermore, we have the following theorem.
Lemma 2.8.Let (I X ,-) be an I-fuzzy topological space and let φ : I X → I be defined by Then φ -is a subbase of one I-fuzzy topology, and denote this I-fuzzy topology by wi(-).wi(-) is called the weakly induced modification of -.
Proof.It is trivial to verify that φ -is a subbase of one I-fuzzy topology.
We now prove that wi(-) is the smallest weakly induced I-fuzzy topology bigger than -.Let - * be any weakly induced I-fuzzy topology on X bigger than -.We need to prove that wi(-) ≤ - * .It suffices to show that φ -(A) ≤ - * (A) for all A ∈ I X .Then it suffices to show that φ -(1 U ) ≤ - * (1 U ) for all U ⊆ X, and this can be obtained by the following computation: thus the conclusion.Proof.The sufficiency is obvious.We need to prove the necessity.It suffices to show that thus the conclusion.
Remark 2.11.From Lemma 2.10 above, we also can get that f → I : (I X ,wi() Hence wi is another functor from I-FTOP to I-WIFTOP.Furthermore, we have the following theorem.
Theorem 2.12.wi is the right adjoint of i.
From Theorems 2.7 and 2.12, we have the main theorem in this paper as follows.Corollary 2.15.Let {(I Xt ,t )} t∈T be a family of I-fuzzy topological spaces and X = t∈T X t .Then wi( t∈Tt ) = t∈T wi(t ).Theorem 2.16.Let {(I Xt ,t )} t∈T be a family of I-fuzzy topological spaces and let different X t s be disjoint.Then wi( t∈Tt ) = t∈T wi(t ).
The readers can easily prove the following theorem.
Theorem 2.17.Let (I X ,-) be an I-fuzzy topological space and let (I Y ,-/ f → I ) be the Ifuzzy quotient space of (I X ,-) with respect to f : X → Y .If (I X ,-) is weakly induced, then (I Y ,-/ f → I ) is weakly induced.

On the relationships between several categories
In Section 2, we study weakly induced modifications of I-fuzzy topologies.Since weakly induced and induced topological spaces play an important role in L-topology, in this section, we will study induced I-fuzzy topologies and the relationships between the categories FYS, I-WIFTOP, I-SFTOP, I-IFTOP, and I-FTOP, where I-IFTOP and I-SFTOP denote the categories of induced I-fuzzy topological spaces and stratified I-fuzzy topological spaces, respectively.In the following discussion, we always assume that I-TOP denotes the category of stratified Chang-Goguen topological spaces.We know that TOP can be regarded as a full (moreover, simultaneously reflective and coreflective) subcategory of I-TOP by Lowen functors.Zhang [16] proved that TOP is a reflective and coreflective full subcategory of FYS and FYS is a reflective and coreflective full subcategory of I-TOP.
From [15], we know that FYS is isomorphic to I-IFTOP.We will prove that I-IFTOP is a reflective and coreflective full subcategory of I-SFTOP and I-IFTOP is a coreflective full subcategory of I-WIFTOP.Let (I X ,-) be an I-fuzzy topological space and let [-] : P(X) → I be defined by [-](U) = -(1 U ) for all U ∈ P(X).Then it is easy to verify that [-] is a fuzzifying topology on X. Definition 3.1 [15].Let (I X ,-) be an I-fuzzy topological space.[-] is called the background topology ofand (X, [-]) is called the background space of (I X ,-).
From the definition above, we get a functor [•] from I-FTOP to FYS.It is easy to verify the following two theorems.
) is a fuzzifying continuous.Theorem 3.3.Let {(I Xt ,t )} t∈T be a family of I-fuzzy topological spaces and let different X t s be disjoint.Then Definition 3.4 [15].Let (I X ,-) be an I-fuzzy topological space on X.If -(A) = r∈I0 -(1 σr (A) ) for all A ∈ I X , then (I X ,-) is called an induced I-fuzzy topological space.If -( λ) = 1 for all λ ∈ I, where λ is the constant function from X to I with value λ, then (X,-) is called a stratified I-fuzzy topological space.Lemma 3.5 [15].Letbe an I-fuzzy topology on X and let φ -: P(X) → I be defined by . Then φ -is the subbase of one fuzzifying topology, and let this fuzzifying topology be denoted by ι(-).
Therefore f → I : (I X ,ω([])) → (I Y ,-) is I-fuzzy continuous.Lemma 3.12.Let (I X ,-) be an I-fuzzy topological space and let (I Y ,) be an induced Ifuzzy topological space.Then f → I : (I Y ,) → (I X ,-) is I-fuzzy continuous if and only if Proof.The sufficiency is obvious.We need to prove the necessity.In fact, we have thus the conclusion.
From Lemmas 3.11 and 3.12, we have the following theorems.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Lemma 2 . 10 .
Let (I Y ,-) be weakly induced and let (I X ,) be an I-fuzzy topological space.Then f → I : (I Y ,-) → (I X ,) is I-fuzzy continuous if and only if f → I : (I Y ,-) → (I X ,wi()) is I-fuzzy continuous.

Theorem 2 . 7 Corollary 2 . 14 .
13. I-WIFTOP is a reflective and coreflective full subcategory of I-FTOP.By the properties of right adjoint, we have the following corollaries.Y.Yue and J. Fang Let (I X ,-) be an I-fuzzy topological space and Y ⊆ X.Then wi(-| Y ) = wi(-) | Y .

Theorem 3 .
13. (1) ω • ι is the right adjoint of the inclusion functor i : I-IFTOP → I-FTOP.(2) ω • [•] is the left adjoint of the inclusion functor i : I-IFTOP → I-SFTOP.Theorem 3.14.I-IFTOP is a reflective and coreflective full subcategory of I-SFTOP and I-IFTOP is a coreflective full subcategory of I-WIFTOP.Hence, I-IFTOP is also a coreflective full subcategory of I-FTOP.Corollary 3.15.FYS is a reflective and coreflective full subcategory of I-SFTOP.Hence TOP is a reflective and coreflective full subcategory of I-SFTOP.

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation