ON GENERALIZED FUZZY STRONGLY SEMICLOSED SETS IN FUZZY TOPOLOGICAL SPACES

We introduce the concept of generalized fuzzy strongly semiclosed, generalized fuzzy almost-strongly semiclosed, generalized fuzzy strongly closed, and generalized fuzzy almost-strongly closed sets. In the light of these deﬁnitions, we also deﬁne some generalizations of fuzzy continuous functions and discuss the relations between these new classes of functions and other fuzzy continuous functions

1. Introduction and preliminaries.Generalized semiclosed (semiopen) and generalized closed (open) sets play an important role in general topology [2,9].Balasubramanian and Sundaram [6] defined generalized fuzzy closed set in fuzzy topological spaces.Later, Abd El-Hakeim [1] introduced the generalized fuzzy semiclosed, generalized fuzzy weakly semiclosed, and generalized fuzzy regular closed sets and studied some of their properties.
In Section 2, we introduce generalized fuzzy strongly semiclosed, generalized fuzzy almost-strongly semiclosed, generalized fuzzy strongly closed, and generalized fuzzy almost-strongly closed sets and establish some of their properties.(We have not seen such discussions on the properties of these sets in general topological spaces.)We also discuss the relations between fuzzy closed sets [3], fuzzy semiclosed sets [3], and fuzzy strongly semiclosed sets [4].
In Section 3, we introduce four new classes of functions among fuzzy topological spaces which are weaker than the classes of fuzzy continuous functions, fuzzy strongly semicontinuous, and fuzzy semicontinuous functions, respectively.(We have not seen corresponding concepts in general topological spaces.)Also, some examples are given, and relationships between these new classes and other classes of fuzzy continuous functions are obtained.
For X, I X denotes the collection of all mappings from X into I = [0, 1].A member λ of I X is called a fuzzy set of X.By (X, τ) or simply by X, we denote a fuzzy topological space (FTS) due to Chang [7].The interior, the closure, and the complement of a fuzzy set µ ∈ I X will be denoted by int µ, clµ, and µ , respectively.Now we introduce some basic notions and results that are used in the sequel.λ ≤ µ and µ is a fuzzy semiopen set [1].
Definition 1.5 (see [8]).Let µ be a fuzzy set in an FTS (X, δ) and define the following fuzzy subsets: is called the fuzzy strong semi-interior of µ, is called the fuzzy strong semiclosure of µ.
Proposition 1.6 (see [8]).Let µ and β be fuzzy sets in an FTS (X, τ).Then the following statements are valid: Every fuzzy open (closed) set is a fuzzy strongly semiopen (semiclosed) set.Every fuzzy strongly semiopen (semiclosed) set is a fuzzy semiopen (semiclosed) set [8].The following examples show that the reverse may not be true in general.

Generalized fuzzy strongly semiclosed sets in fuzzy topological spaces
and τ = {0, 1,µ}.It is easy to see that λ is a gf-closed set but it is neither gfsts-closed nor fuzzy closed.
However, the intersection of two gfst-semiclosed sets is not a gfst-semiclosed set.We can see this in the following example.

Theorem 2.13. Let (X, τ) be a fuzzy topological space. A fuzzy set µ ∈ I X is a gfastsemiopen (resp., gfst-open, gfast-open) set if and only if
Proof.The proof is similar to the proof of Theorem 2.12.

Generalized fuzzy strongly semicontinuous functions.
In this section, four new classes of functions are introduced.Their relationships with other fuzzy continuous functions are established.
It is clear that f is gfst-semicontinuous but not fuzzy continuous.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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Definition 3 . 1 . 1 )Example 3 . 2 .
Let (X, τ) and (Y , φ) be two fuzzy topological spaces.A mapping f : (X, τ) → (Y , φ) is called (a) generalized fuzzy strongly semicontinuous (gfst-semicontinuous) if the inverse image of every fuzzy closed set in Y is a gfst-semiclosed set in X; (b) generalized fuzzy almost-strongly semicontinuous (gfast-semicontinuous) if the inverse image of every fuzzy closed set in Y is a gfast-semiclosed set in X; (c) generalized fuzzy strongly-continuous (gfst-continuous) if the inverse image of every fuzzy closed set in Y is a gfst-closed set in X; (d) generalized fuzzy almost-strongly continuous (gfast-continuous) if the inverse image of every fuzzy closed in Y is a gfast-closed set in X.Thus we have the following diagram: None of these implications are reversible, as the following counterexamples state.Let X = {a, b, c}, Y = {p, q}.Define τ = {0, 1,λ}, and φ = {0, 1,µ} are FTS on X and Y , respectively, where λ ∈ I X is such that λ(a) = 0, λ(b) = λ(c) = 0.5, and µ