FUZZY cγ-OPEN SETS AND FUZZY cγ-CONTINUITY IN FUZZIFYING TOPOLOGY

The concepts of fuzzy cγ-open sets and fuzzy cγ-continuity are introduced and studied in fuzzifying topology and by making use of these concepts, some decompositions of fuzzy continuity are introduced.

where Ᏺ(X) is the family of all fuzzy sets in X.
(2) From Lemma 3.2, we have Proof.From Theorem 3.3 the proof is obtained.Theorem 3.5.Let (X, τ) be a fuzzifying topological space, then ( Proof.From the properties of the interior and closure operations and [9, Theorem 2.2(3)], (1) (a) (2) The proof is obtained from (1).Remark 3.6.In crisp setting, that is, in case that the underlying fuzzifying topology is the ordinary topology, we have Of course the implication "→" in (3.2) is either the Lukaciewicz's implication or the Boolean's implication since these implications are identical in crisp setting.But in fuzzifying setting the statement (3.2) may not be true as illustrated by the following counterexample.

where Ᏺ N (P (X)) is the set of all normal fuzzy subsets of P (X), has the following properties:
(1) 2) and ( 3), then cγN assigns a fuzzifying topology on X which is denoted by τ cγN ∈ Ᏺ(P (X)) and defined as (2) The proof is immediate.
Remark 7.2.In crisp setting, that is, in the case that the underlying fuzzifying topology is the ordinary topology, one can have But this statement may not be true in general in fuzzifying topology as illustrated by the following counterexample.
Counterexample 7.3.Let (X, τ) be the fuzzifying topological space defined in Counterexample 3.7.Consider the identity function f from (X, τ) onto (X, σ ), where σ is a fuzzifying topology on X defined as follows: Theorem 7.4.Let (X, τ) and (Y , U ) be two fuzzifying topological spaces.For any and so the result holds.

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

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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