ON FUZZY FUNCTION SPACES

In [3], we started the investigation of compactness in fuzzy function spaces in FCS, the category of fuzzy convergence spaces as defined by Lowen/Lowen/Wuyts [8]. This paper goes somewhat deeper in the investigation of fuzzy function spaces using the notion of splitting and conjoining structures on fuzzy subsets. We discuss the connection to the exponential law and give several examples of such structures. As a special case, we study a notion of fuzzy compact open topology.


Introduction.
The theory of fuzzy topological spaces is meanwhile highly developed.Especially, compactness and separation axioms have been thoroughly studied.Yet, one important field of classical topology has not yet attained wide attention in fuzzy topology: the theory of function spaces.Function spaces play an important role in functional analysis, in the theory of differential equations, in complex analysis, and in almost every other branch of modern mathematics, not to forget in topology itself.Therefore, it seems desirable to study function spaces also in fuzzy topology.In the meantime, three papers on this subject have appeared (Peng [11], Dang and Behera [2], and Alderton [1]).The fact that FTS, the category of fuzzy topological spaces (Lowen [9]), is not cartesian closed led Lowen and Lowen [7] to the definition of FCS, the category of fuzzy convergence spaces.This paper takes FCS as a starting point to discuss certain fuzzy function space structures via splitting and conjoining structures.It continues a previous paper by the author, where compactness in fuzzy function spaces in FCS was considered [3] and also considers function spaces in FTS.

Preliminaries.
Let X be a nonvoid set.Fuzzy subsets of X are denoted by A, B, C,... ∈ [0, 1] X , (ordinary) subsets of X are denoted by small italics a,b,c,... ⊂ X.For a ⊂ X, we denote by 1 a the characteristic function of a and in case a = {x}, we write 1 x .For the characteristic function of the whole set X, Y , Z, we write for short again X, Y , Z.The fundamental definitions of fuzzy set theory and fuzzy topology are assumed to be familiar to the reader.We especially take Lowen's definition of fuzzy topology [9].In order to make this paper self-contained, however, we summarize the main results of our papers [3,4,5].Given a fuzzy subset A ∈ [0, 1] X , we denote F X (A) := {B ∈ [0, 1] X : B ⊂ A} and call A 0 := {x ∈ X : A(x) > 0} the support of A. For A ∈ [0, 1] X , we call a fuzzy subset of the form B = A ∩ 1 B 0 a crisp fuzzy subset of A.
If A = X, we regain the usual definition.
For a function f : X → Y , we define its restriction on A by f | A(D) := f (D) (D ∈ F X (A)) and the corresponding inverse image by (f A nonempty collection F ⊂ F X (A) is called a fuzzy filter on A if and only if it does not contain the empty fuzzy set ∅ := 1 ∅ , is closed under finite intersections, and contains, for F ∈ F, every fuzzy superset A ⊃ G ⊃ F .B ⊂ F X (A) is called a fuzzy filter basis on A if and only if it is not empty, does not contain the empty fuzzy set, and the intersection of two of its members contains a member of B. For a fuzzy filter basis are prime fuzzy filters.The set F(A) of fuzzy filters on A is ordered by set inclusion.For F ∈ F(A), the set P(F) of all prime fuzzy filters finer than F is inductive and, by Zorn's lemma, there exist minimal elements in P(F), the set of which is denoted by P m (F) (cf.Lowen [10]).For a fuzzy filter F ∈ F(A), the system ı(F) := {F 0 : F ∈ F} is a filter on A 0 .F is a prime fuzzy filter if and only if ı(F) is an ultrafilter.We further call for a fuzzy filter its characteristic value (Lowen/Lowen [6]).
For A ∈ [0, 1] X , we call a mapping lim : a fuzzy convergence on A if and only if the following conditions are satisfied: (cf. [5,7,8]).The pair (A, lim) is then called a fuzzy convergence space (fcs for short).A fuzzy topological space (X, ∆) can be considered as an fcs if we put, for F ∈ F(X), which is the definition of limit of a fuzzy filter due to Lowen [10].
For two fuzzy convergences lim, lim on the same fuzzy set A ∈ [0, 1] X , we say that lim is finer than lim if and only if, for every prime fuzzy filter F ∈ F(A), we have lim F ⊂ lim F. We then write lim ≤ lim .It is easily verified that, for two fuzzy topologies Γ , ∆ on X, we have Γ ≤ ∆ (i.e., Γ ⊂ ∆) if and only if lim(Γ ) ≤ lim(∆).

Continuous convergence on fuzzy subsets (The space C(A, B)). Let in this number
We then have ⊂ B}, i.e., we do not need to distinguish between mappings from A 0 to B 0 and fuzzy mappings from A to B. For a mapping g : A 0 → B 0 , we put i.e., B A = Π x∈A 0 B (identifying g and (g(x)) x∈A 0 in the natural way) is a fuzzy set on B A 0 0 .We consider the evaluation map ev : ) is assumed, then ev : B A × A → B is a fuzzy mapping [3].We further denote by π B A : B A × A → B A the restriction of the projection pr g and by π A : The fuzzy convergence structure of continuous convergence on B A is defined as follows (cf.[3,7,8]).For (A, lim A ), (B, lim B ) fcs's, g : A → B a fuzzy mapping and F ∈ F(B A ) a prime fuzzy filter, we put For an arbitrary fuzzy filter F ∈ F(B A ), we derive c-lim F by (PST).c-lim then satisfies (PST), (F1p), and (F2p) and, in general, fails to satisfy (C1).Therefore, we speak of c-lim as a " weak fuzzy convergence structure" (cf.[3]).
If M ⊂ B A is a fuzzy subset of B A , we call c-lim | M the (weak) fuzzy convergence structure of continuous convergence on M and denote this (weak) fuzzy convergence again by c-lim.The next proposition shows that we can calculate the fuzzy convergence of continuous convergence for certain M ⊂ B A "from inside."Proposition 3.1.Let (A, lim A ), (B, lim B ) be fcs's and M = B A ∩1 M 0 be a crisp fuzzy subset of B A .If we put for a prime fuzzy filter here, the fuzzy functions on the left sides are defined on B A × A and the ones on the right sides are defined on M × A).Hence, we get

Conversely, let α ∈ C M (F,g) and Θ ∈ F(B A × A) be a prime fuzzy filter such that π B A (Θ) ≤ [F] and let x ∈ A 0 . Then, as ı(π B A (Θ)) = ı([F]
) and M is a crisp fuzzy subset, we get that M ∈ π B A (Θ) and, therefore, also (i) ev : Proof.Using Proposition 3.1, we can copy the corresponding proof of [3,Prop. 4.6].
We now put for two fcs's (A, lim A ) and (B, lim B ) and define the fuzzy subset we showed that, for a continuous fuzzy mapping g : (A, lim A ) → (B, lim B ) and for 0 < α ≤ η(g), we have α1 g ⊂ c-lim[α1 g ].Hence, (C(A, B), c-lim) satisfies the axiom (C1), i.e., is an fcs.We finally mention a result due to Lowen/Lowen [7].Let X, Y , Z be nonvoid sets and f : X × Y → Z be a mapping.We define a mapping ϕ(f ) : The just-defined bijection ϕ : Z X×Y → (Z Y ) X is called an "exponential map" (Poppe [12]). (3.9) Here, C(Y , Z) is provided with the fuzzy convergence of continuous convergence and X × Y with the product fuzzy convergence lim X × lim Y .
The reverse inclusion follows using the continuity of the evaluation map (Proposition 3.2) and the continuity of the fuzzy product-mapping (Proposition 2.2) in exactly the same way as the proof of the corresponding "classical" theorem 2.2, (a)⇒(b) in Poppe [12].

Splitting and conjoining fuzzy convergences
Definition 4.1.Let (A, lim A ), (B, lim B ) be fcs's, M ⊂ B A and lim be a fuzzy convergence on M.
(i) lim is called conjoining for M if and only if c-lim ≤ lim holds on M.
(ii) lim is called splitting for M if and only if lim ≤ c-lim holds on M.
As an immediate consequence of Proposition 3.2, we obtain the following proposition.
Proposition 4.2.Let (A, lim A ), (B, lim B ) be fcs's and M ⊂ B A .The following hold: (i) If lim is conjoining for M, then ev : a crisp fuzzy subset, then, from the continuity of ev : (M × A, lim × lim A ) → (B, lim B ), we get that lim is conjoining.

Proposition 4.3. Let (Y , lim Y ), (Z, lim Z ) be fcs's and lim be a fuzzy convergence for C(Y , Z). Then the following are equivalent:
(i) lim is splitting for C(Y , Z), (ii) for each fcs (X, lim X ), we have Proof.Let first lim be splitting for C(Y , Z).From Proposition 3.3, we get that, for a continuous fuzzy mapping f : X × Y → Z, the fuzzy mapping ϕ(f ) : X → (C(Y , Z), c-lim) is continuous.Hence, for a prime fuzzy filter F ∈ F(X), we have i.e., ϕ(f ) is (lim X , lim)-continuous.(i) lim is conjoining for C(Y , Z), (ii) for each fcs (X, lim X ), we have Proof.Let first lim be conjoining for C(Y , Z), and let f : (X, lim X ) → (C(Y , Z), lim) be continuous.As c-lim ≤ lim, then also f : (X, lim X ) → (C(Y , Z), c-lim) is continuous.

Examples for splitting and conjoining fuzzy convergences
5.1.The discrete and the indiscrete fuzzy convergences.Let A ∈ [0, 1] X .If we put for a prime fuzzy filter F ∈ F(A) and derive lim ı respectively lim δ for arbitrary fuzzy filters on A by (PST), then the following proposition holds.
Proof.(i) That lim ı is a fuzzy convergence on A is obvious and, obviously, lim δ satisfies axioms (F1p), (PST), and (C1).If F, G ∈ F(A) are prime fuzzy filters and and also (F2p) holds.
lim ı is called the indiscrete fuzzy convergence on A and lim δ is called the discrete fuzzy convergence on A. A, B), then the discrete fuzzy convergence on M is conjoining for M.

The fuzzy convergence of pointwise convergence.
Let (A, lim A ), (B, lim B ) be fcs's.We define a fuzzy convergence on B A by putting, for a prime fuzzy filter and derive p-lim F for an arbitrary fuzzy filter on B A by (PST), i.e., p-lim is the initial fuzzy convergence on B A respectively the (π x ,B) x∈A 0 .p-lim is called the fuzzy convergence of pointwise convergence on B A [3].In [3], we proved the following proposition.
Y be constant fuzzy sets of the same height α > 0 and (A, lim A ), (B, lim B ) be fcs's.Then p -lim ≤ c-lim.
Corollary 5.4.Under the assumptions of Proposition 5.3, the fuzzy convergence of pointwise convergence p -lim is splitting for B A .

The fuzzy convergence of strictly continuous convergence.
Let (A, lim A ), (B, lim B ) be fcs's and M ⊂ B A .We put for a prime fuzzy filter F ∈ F(M) and a fuzzy mapping g : and derive sc-lim F for arbitrary fuzzy filters on M by (PST).
Proposition 5.5.Under the assumptions mentioned above, sc -lim is a fuzzy convergence on M.

Splitting and conjoining fuzzy topologies.
In this number, we only consider fuzzy topological spaces (X, ∆), (Y , Γ ) in the sense of Lowen [9] and crisp subspaces of Y X .As in the (classical) theory of convergence space (Poppe [12]), we call a fuzzy topology T on a crisp subset M = 1 M 0 ⊂ Y X splitting (resp.conjoining) for M if and only if the corresponding fuzzy convergence lim(T) is splitting (resp.conjoining) for M.
6.1.The discrete and the indiscrete fuzzy topologies.Let X be a nonvoid set.If we put ∆ δ := F X (X) and ∆ ı := {α : 0 ≤ α ≤ 1}, then, obviously, ∆ δ and ∆ ı are fuzzy topologies on X. ∆ δ is called discrete fuzzy topology on X and ∆ ı is called the indiscrete fuzzy topology on X. (ii) If M ⊂ C(X, Y ), then the discrete fuzzy topology is conjoining for M.

The fuzzy compact open topology.
In [11], Peng defines a notion of fuzzy compact open topology using the notion of N-compactness due to Wang [13].Here, we alter his definition slightly using Lowen's [9] definition of compactness.Let (X, ∆), (Y , Γ ) be fuzzy topological spaces.For a subset k ⊂ X and G ∈ [0, 1] Y , we put It is easily verified that the system is a subbase for a fuzzy topology on Y X , the fuzzy compact open topology ∆ co on Y X .If we restrict the functions (1 k ,G) on C(X, Y ), then the subspace topology of ∆ co is also denoted by ∆ co .
For the proof of the next proposition, we need two lemmas, the proofs of which are left to the reader.Lemma 1.Let X be a nonvoid set and ∆, Γ be fuzzy topologies on X. If, for every D ∈ Γ and for every x ∈ X such that D(x) =: α x > 0, there is Lemma 2. Let (X, ∆) and (Y , Γ ) be fuzzy topological spaces and let (X × Y ,∆ × Γ ) be their product space.Then G ∈ ∆ × Γ if and only if, for every (x, y) ∈ X × Y and every > 0, there are fuzzy sets H x ∈ ∆, K y ∈ Γ such that H x × K y ⊂ G and H x × K y (x, y) ≥ G(x, y) − .Proposition 6.3.Let (X, ∆) and (Y , Γ ) be fuzzy topological spaces and T be a conjoining fuzzy topology on Y X (respectively C(X, Y )).Then T ≥ ∆ co .
As T is conjoining, the evaluation map is continuous (Proposition 4.2 and [5, Prop.4.3]) and, hence, ev −1 (G) ∈ T ×∆.Let > 0. Lemma 2 yields the existence of fuzzy sets From this, we see that the system {O x : x ∈ k} is an open cover of (α g − ( /2))1 k .As 1 k is compact, there are finitely many x 1 ,...,x n ∈ k such that The arbitrariness of δ > 0 now yields the assumption.

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.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: lim) fcs, and ı B := id X | B : B → A is the fuzzy inclusion, we call lim | B := init(lim,ı B ) the fuzzy convergence on B induced by lim and the pair (B, lim | B ) a fuzzy subspace of (A, lim).We have lim | B F = B ∩ lim[F] for a prime fuzzy filter F ∈ F(B).

Proposition 2 . 1 .
are fuzzy mappings and if we define, as usual, the product-mapping f × k(a, c) := (f (a), k(c)), then it is easily verified that f × k(A × C) ⊂ B × D. Hence, we can define the fuzzy product-mapping g × h := f × k | A × C : A × C → B × D. The simple proofs of the next two propositions are left to the reader.Let, in the situation above, F ⊂ A × C. If π A respectively π C , are the fuzzy projections from A × C to A respectively C, and π B respectively π D , are the fuzzy projections from B × D to B respectively D, then g and ev(Θ ) = ev(Θ) (where again the fuzzy functions on the left sides are defined on M × A and those on the right sides are defined on B A × A), this yields α ∈ C([F], g) which completes the proof.Proposition 3.2.Let (A, lim A ), (B, lim B ) be fcs's and M = B A ∩1 M 0 be a crisp fuzzy subset of B A .The following hold: