Maximization based topologies and their relation with Γ-convergence of intuitionistic fuzzy sets

1 Introduction

General Topology has turn out to be one of the essential ingredients of mathematics. As a result of a concrete research activities in these days, this subject has been revealed to be valuable in modeling, numerous problems which come up in some branches of practical sciences as Computer Science, Economics and Artificial Intelligence, etc. The study of Γ-convergence is started with Wijsman [21], when the infimal convergence of convex functions was defined. Later on the formal definition of Γ-convergence is given by De Giorgi and Franzoni in the context of calculus of variation [11, 12]. A comprehensive and organized handling of the general theory of Γ-convergence as well as applications in control theory, convex analysis, game theory and optimal design problems, etc., can be found in [3, 8, 16–18] and references therein. For more recent advances, Ferreira and Gomes [13] studied the long-term convergence of finite state mean-field games using Γ-convergence, Öztürk et al. [19] proposed the notion of strong Γ-ideal convergence in probabilistic norm spaces, Bonder et al. [7] investigated the critical Sobolev embedding for variable exponent Sobolev spaces from the point of view of Γ-convergence and Briane et al. [9] studied the Γ-convergence of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients.

Fuzzy mathematics has confirmed its effectiveness over the years and capable to resolve lot of troubles which traditional logic incapable to grip. In the framework of fuzzy set theory Γ^--convergence (resp. Γ⁺-convergence) of fuzzy sets are lower (resp. upper) Kuratowski-Painlevé convergence of their endographs, whereas if epigraphs of fuzzy sets are used in place of endographs we have Epi-convergence. Which is dual to Γ-convergence. Basically it is dependent upon the nature of problem, for problems related to maximization Γ-convergece and for problems related to minimization Epi-convergence serves the best. Greco [14, 15] used Γ-convergence in his investigation of compactness in metric space of upper semicontinuous, compact-supported, normal fuzzy subsets. Pedraza et al. [20] gave a set theoretic characterization for convergence of fuzzy sets in supremum metric and they also discussed relation between Γ-convergence, uniform convergence and convergence in supremum metric.

Atanassov [1] gave the notion of intuitionistic fuzzy set (IFS), which is an extension of Zadeh’s fuzzy set [22]. It enriches fuzzy set theory with a notion of indeterminacy expressing hesitation or abstention. IFS deals with more complicate and ambiguous real life problems, allowing more freedom. It is a function f from a topological space (X, τ) to 𝕋 , where 𝕋 = { ( α , β ) ∈ [ 0 , 1 ] 2 : α + β ≤ 1 } . We refer to the book of Atanassov [2] for the basics on IFS. It has received more attention and has been applied not only in theoretical fields (differential equation, variational analysis, fixed point theory, etc.) but also in practical fields (artificial intelligence, engineering, medical, similarity measure, decision making, etc.). Intuitionistic fuzzy measure was developed by Ban in [4] and successfully used in several applications. Atanassov [2] in his book gave the detailed survey on the theory and applications of intuitionistic fuzzy set and also described new directions and research problems. Let IF ( X ) and USC ( X ) be the sets of all IFSs and all upper semicontinuous IFSs over the topological space (X, τ _X ), respectively.

In [5] Γ-convergence for IFS is defined and studied. In this paper, with the help of Maxima operator three topologies τ M - , τ M + and τ_M are defined on IF ( X ) . It is proved that the topologies ( IF ( X ) , τ M - ) , ( IF ( X ) , τ M + ) and ( IF ( X ) , τ M ) are compact topological spaces. Classically the relation between minimization based topologies and Epi-convergence has been studied [17] but only for first countable spaces. Our aim is to establish the connection between maximization based topologies and Γ-convergence at the level of nets and in generalised setting of intuitionistic fuzzy set theory. Γ^--convergence is compatible with convergence in topology τ M - , but same is not true for Γ⁺ and τ M + in general. However, we characterize Γ⁺ and Γ convergence in τ M + and τ_M when IF ( X ) is restricted to USC ( X ) and topological space (X, τ _X ) is locally compact Hausdroff. As the main result of our investigations, we prove (among others) the following result:

Theorem. Consider (X, τ _X ) is locally compact space. Then a net {f _λ } _λ∈Λ in USC ( X ) is

Γ^--convergence to f if and only if it is convergent to f in topological space ( USC ( X ) , τ M - ) ;

Plan of the paper is as follows. Section 2 is about basic definitions and results. In Section 3 definition and some applications of Γ-convergence are presented. Section 4 is dedicated for the study of topologies based on maxima operator and in Section 5 their relation is discussed with Γ-convergence.

2 Basic concepts

At first let us recall some of the definitions and necessary topological structures on 𝕋 from [5]. The following order is used in 𝕋 : for ζ = ( α ζ , β ζ ) , η = ( α η , β η ) ∈ 𝕋 , ζ ≤ η if and only if α _ζ  ≤ α _η and β _ζ  ≥ β _η , also ζ < η if and only if either α _ζ  < α _η and β _ζ  ≥ β _η or α _ζ  = α _η and β _ζ  > β _η . With this ordering 𝕋 is a complete lattice, but it is not a totaly ordered set. The supremum and infimum are sup(ζ, η) = (max(α _ζ , α _η ), min(β _ζ , β _η )) and inf(ζ, η) = (min(α _ζ , α _η ), max(β _ζ , β _η )), respectively. For a subset A of 𝕋 , sup A : = (sup α _x , inf β _x ) and inf A : = (inf α _x , sup β _x ) for all x ∈ A. Note that (1, 0) is the largest and (0, 1) is the smallest element in 𝕋 .

In this paper, d _e is used for usual Euclidean metric on ℝ 2 or any of its subsets. For a metric space (X, d) we denote B _d  (x, r) as an open ball for radius r > 0 and for A ⊆ X define B _d  (A, r) = ∪ _x∈AB _d  (x, r). In case of topological space ( X , τ ) , N x is the set of all neighborhoods of x ∈ X .

For a net {x _λ } _λ∈Λ in 𝕋 ,

the lower limit of {x _λ } _λ∈Λ as lim inf λ ∈ Λ x λ = sup λ ′ ∈ Λ inf λ ≥ λ ′ x λ ;

Definition 2.1. A net {x _λ } _λ∈Λ in 𝕋 is:

Lower convergent to x ∈ 𝕋 if lim inf λ ∈ Λ x λ ≥ x , C _lower is the collection of all such x;

The order ≤ is quite useful because in this ordering the lower and the upper convergences are topological. Consider the lower topology induces by lower pseudometric d _lower  (ζ, η) = max(α _ζ  - α _η , β _η  - β _ζ , 0), then it is easy to verify that convergence in this topology and lower convergence coincides. Similarly, for the upper convergence we have upper topology with the upper pseudometric d _upper  (ζ, η) = max(α _η  - α _ζ , β _ζ  - β _η , 0).

Let f ∈ IF ( X ) then the endograph or hypograph of f is defined as end f : = { ( x , ζ ) ∈ X × 𝕋 : ζ ≤ f ( x ) } , similarly epigraph is defined as epi f : = { ( x , ζ ) ∈ X × 𝕋 : f ( x ) ≤ ζ } and the ζ-cut is defined as [f]  ^ζ  : = {x ∈ X : ζ ≤ f (x)}. A function can described uniquely by its endograph or ζ-cuts. We will consider the endographs in the topological space ( X × 𝕋 , τ × τ d e ) . The support of f is defined as Supp ( f ) = ⋃ ( 0 , 1 ) < ζ ∈ 𝕋 [ f ] ζ ¯ .

Definition 2.2. For f ∈ IF ( X ) we define:

lower limit at x ∈ X as f _ ( x ) = lim inf y → x f ( y ) = sup U ∈ N x inf y ∈ U f ( y ) ;

Definition 2.3. An IFS f ∈ IF ( X ) is called:

lower semicontinuous at x ∈ X if f ( x ) ≤ f _ ( x ) ;

If (X, d) is a metric space then f is upper (resp. lower) semicontinuous for all x ∈ X if and only if f is continuous at x when 𝕋 is endowed with upper (resp. lower) quasi-pseudometric. Consequently f is upper (resp. lower) semicontinuous at x if for every ɛ > 0 we can find δ _ɛ such that d _upper  (f (x), f (y)) < ɛ (resp. d _lower  (f (x), f (y)) < ɛ) for all y ∈ B _d  (x, δ _ɛ ) .

Consider a net {A _λ } _λ∈Λ in topological space (X, τ) then

a subset of Λ is said to be residual if it contains all indices at or beyond some index λ;

For more details see [6, page 2, 145].

Definition 2.4. A net {A _λ } _λ∈Λ in (X, τ) is:

lower Kuratowski-Painlevé convergent to A ⊆ X if A ⊆ LiA _λ ;

At the end of this section, we recall the definition of maxima operator that is central in our study as M A ( f ) = sup x ∈ A f ( x ) , where A is any subset of X and f ∈ IF ( X ) with the convention that suprema is (0, 1) for empty set. Note that M _A  (f) is the least upper bound of set f (A) in 𝕋 .

3 Γ-convergence

The Γ-convergence is generally known as a notion of convergence in variational analysis. However, it can play an important role to solve problems of more vague nature described in framework of intuitionistic fuzzy set theory. At first definition of Γ-convergence for IFS is given and then their main properties are discussed.

Definition 3.1. [5] A net {f _λ } _λ∈Λ in IF ( X ) , is:

Γ^--convergent to f if the net {endf _λ } _λ∈Λ is lower Kuratowski-Painlevé convergent to endf i.e. endf ⊆ Liendf _λ ;

In the same way, epi convergence is defined if in above definition endographs is replaced by epigraphs.

Note that endographs are considered in ( X × 𝕋 , τ X × τ d e ) . To work on Γ-convergence, there are some useful functions defined as: ( Li f λ ) ( x ) = inf U ∈ N x lim inf λ ∈ Λ M U ( f λ )(3.1) and ( Ls f λ ) ( x ) = inf U ∈ N x lim sup λ ∈ Λ M U ( f λ ) .(3.2)

If (X, d) is a metric space then there are the following simplifications ( Li f λ ) ( x ) = inf ɛ > 0 lim inf λ ∈ Λ M B d ( x , ɛ ) ( f λ ) ;(3.3) ( Ls f λ ) ( x ) = inf ɛ > 0 lim sup λ ∈ Λ M B d ( x , ɛ ) ( f λ ) .(3.4)

The following lemma will show the importance of above functions.

Lemma 3.2. [5, Lemma 4.4] For a net {f _λ } _λ∈Λ in IF ( X ) . Then Li end f λ = end Li f λ and Ls end f λ = end Ls f λ .

Hence Γ-convergence can be completely described by the functions Lif _λ and Lsf _λ and it is not hard to prove that these functions are upper semicontinuous.

Proposition 3.3. For {f _λ } _λ∈Λ in IF ( X ) and U be any open subset of X . Then

M U ( Li f λ ) ≤ lim inf λ ∈ Λ M U ( f λ ) ;

Proof. They follow immediately from the Equations (3.1) and (3.2), respectively. ■

Proposition 3.4. Take any compact subset K in X and {f _λ } _λ∈Λ in IF ( X ) . Then M K ( Ls f λ ) ≥ lim sup λ ∈ Λ M K ( f λ ) .

Proof. There exists a sub net { f λ ´ } λ ´ ∈ Λ ´ of {f _λ } _λ∈Λ such that lim λ ´ ∈ Λ ´ M K ( f λ ´ ) = lim sup λ ∈ Λ M K ( f λ ) .

Also, we can find a net { x λ ´ } λ ´ ∈ Λ ´ in K for which lim λ ´ ∈ Λ ´ f λ ´ ( x λ ´ ) = lim λ ´ ∈ Λ ´ M K ( f λ ´ ) .

Since K is compact there will be a cluster point x ∈ K of { x λ ´ } λ ´ ∈ Λ ´ . Take U ∈ N x , then lim sup λ ∈ Λ M U ( f λ ) ≥ lim sup λ ´ ∈ Λ ´ M U ( f λ ´ ) ≥ lim λ ´ ∈ Λ ´ f λ ´ ( x λ ´ ) = lim sup λ ∈ Λ M K ( f λ ) .

Hence, taking infimum on all U ∈ N x results into ( Ls f λ ) ( x ) ≥ lim sup λ ∈ Λ M K ( f λ ) . ■

The similar inequality M K ( Li f λ ) ≥ lim inf λ ∈ Λ M K ( f λ ) is not true in general as the following example shows.

Example 3.5. Consider a sequence of IFS over [0, 1] defined as f n ( x ) = { ( x 2 2 , x 2 2 ) , n is even ; ( 1 - x 2 2 , 1 - x 2 2 ) , n is odd . Then ( Li f n ) ( x ) = ( min ( x 2 , 1 - x 2 ) , max ( x 2 , 1 - x 2 ) ) and ( Ls f n ) ( x ) = ( max ( x 2 , 1 - x 2 ) , min ( x 2 , 1 - x 2 ) ) .

Also M_[0,1] (Lif _n ) = (1/4, 1/4), where as lim inf n ∈ ℕ M [ 0 , 1 ] ( f n ) = ( 1 / 2 , 0 ) .

In [5, Theorem 4.5], it is proved that lower pointwise convergence implies Γ^--convergence and Γ⁺-convergence implies upper pointwise convergence. These implications cannot be reversed in general, see [5, Example 4.6]. Therefore, in general we do not have any relation between Γ-convergence and pointwise convergence.

The main idea for defining the topology for fuzzy sets is to use the maxima operator M defined on the IF ( X ) or USC ( X ) .

Definition 4.1. Consider the set IF ( X ) and operator M _A  (f) on it. Then

τ M - is defined to be the weakest topology on IF ( X ) for which the operator M _U  (f) is lower semicontinuous for any open subset U of X;

To work on these topologies consider the subsets of IF ( X ) , A U , ζ : = { f ∈ IF ( X ) : M U ( f ) > ζ } and B K , ζ : = { f ∈ IF ( X ) : M K ( f ) < ζ } , where U ∈ τ _X , K is compact subset of X and ζ ∈ 𝕋 .

Also, Let A be the collection of all such A_U,ζ and {f _l }, where f _l  (x) = (0, 1) for all x ∈ X. Similarly B is the collection of all such B_K,ζ and {f _u }, where f _u  (x) = (1, 0) for all x ∈ X.

Proposition 4.2. Consider the above given collections A and B. Then

A is subbase for the topology ( IF ( X ) , τ M - ) ;

Proof. First of all it is clear that indeed A and B are subbasis, we only need to show that the respective topologies are τ M - and τ M + .

(1): First we show that M _U  (f) is lower semicontinuous in topology induced by A and then it will showed to be the weakest topology. Let N f be the set of all neighbourhoods of f ∈ IF ( X ) and U ∈ τ _X , then lim inf g → f M U ( g ) = sup V ∈ N f inf g ∈ V M U ( g ) .

Also for g ∈ V there will be ζ ∈ 𝕋 such that M _U  (g) > ζ ≥ M _U  (f), so lim inf g → f M U ( g ) ≥ M U ( f ) .

Let τ′ be a topology in which M _U  (f) is lower semicontinuous for any U ∈ τ _X . Take A_U,ζ ∈ A. Since M _U is lower semicontinuous, for any f ∈ A_U,ζ, thus sup V ′ ∈ N f ′ inf g ∈ V ′ M U ( g ) ≥ M U ( f ) > ζ , where N f ′ ⊆ τ ′ is the set of all neighbourhoods of f. Hence there will be V ′ ∈ N f ′ such that for all g ∈ V′ M _U  (g) > ζ, so A_U,ζ ∈ τ′.

(2): Analogous to (1).

(3): Follows from (1) and (2), because τ_M is the weakest topology stronger than both τ M - and τ M + . ■

Remark 4.3. For a net {f _λ } _λ∈Λ in IF ( X ) , it is

convergent to f ∈ IF ( X ) in the topology τ M - if and only if M U ( f ) ≤ lim inf λ ∈ Λ M U ( f λ ) ;(4.1)

The convergence in τ M - implies Inequality 4.1 because M _U  (f) is lower semicontinuous in τ M - , where as Inequality 4.2 is implied from convergence in τ M + due to the fact that M _K  (f) is upper semicontinuous in τ M + . Also in above, the reverse implications are consequences of Proposition 4.2.

These topologies, also make IF ( X ) compact. To prove this fact first, the famous Alexander lemma for compactness related to subbase is stated here.

Lemma 4.4. [10, Lemma 4.4.4] Let Δ be a subbase for a topological space (X, τ _X ). If every cover of X by members of Δ has a finite subcover then X is compact.

This lemma is useful to prove the following theorem.

Theorem 4.5. The topological spaces ( IF ( X ) , τ M - ) , ( IF ( X ) , τ M + ) and ( IF ( X ) , τ M ) are compact.

Proof. Since τ_M is stronger than τ M - and τ M + , it is sufficient to prove compactness of ( IF ( X ) , τ M ) . Take a cover IF ( X ) = ⋃ i ∈ Δ 1 A U i , ζ i ∪ ⋃ j ∈ Δ 2 B K j , ζ j .

Let m ( x ) : = inf { ζ i : i ∈ Δ 1 , x ∈ U i } , with the convention that infimum of empty set is (1, 0). It is clear that for x ∈ U _i we have m (x) ≤ ζ _i for all i ∈ Δ₁. Therefore m ∉ ⋃ _i∈Δ1A_{U i ,ζ i} , so there will be some j ∈ Δ₂ such that m ∈ B_{K j ,ζ j} , for simplicity, we fixed these K _j  = K and ζ _j  = ζ.

For any x ∈ K m (x) < ζ, thus by definition of m there is some i _x in Δ₁ such that x ∈ U _{i x} and ζ _{i x}  < ζ. This means K ⊆ ⋃ x ∈ K U i x .

But K is compact there will be x₁, x₂, …, x _n such that K ⊆ ⋃ l = 1 n U i x l .(4.3)

Take f ∈ IF ( X ) if M _K  (f) < ζ then f ∈ B_K,ζ. Otherwise when M _K  (f) ≥ ζ, due to ζ > ζ _{i x l} there is x ∈ K such that f (x) > ζ _{i x l} for all l = 1, 2, . . . , n. But Inclusion 4.3 implies that x is in some U _{i x l} . Hence, f ∈ A_{U i x l ,ζ i x l} and finally IF ( X ) = ⋃ l = 1 n A U i x l , ζ i x l ∪ B K , ζ . ■

In comparison between Γ-convergence and convergence in τ_M, it was found that Γ-convergence implies convergence in τ_M.

Proposition 4.6. Let {f _λ } _λ∈Λ be a net in IF ( X ) . If {f _λ } _λ∈Λ is Γ-convergent to f then it is convergent to f in the topology τ_M.

Proof. It follows from Propositions 3.3, 3.4 and Remark 4.3. ■

Now Γ^--convergence and Γ⁺- convergence are discussed separately. Γ^--convergence is topological as the following proposition states.

Proposition 4.7. A net {f _λ } _λ∈Λ in IF ( X ) is Γ^--convergent to f if and only if it is convergent to f in the topology τ M - .

Proof. Γ^--convergence implies convergence in τ M - , it follows from Proposition 3.3 and Remark 4.3.

Conversely, assume that {f _λ } _λ∈Λ is convergent to f in the topology τ M - . So M U ( f ) ≤ lim inf λ ∈ Λ M U ( f λ ) , where U ∈ τ X .

Therefore ( Li f λ ) ( x ) = inf U ∈ N x lim sup λ ∈ Λ M U ( f λ ) ≥ inf U ∈ N x M U ( f ) ≥ f ( x ) .

Hence, endf ⊆ endLif _λ . ■

There is no hope to prove a similar relation between Γ⁺-convergence and convergence in the topology τ M + . Because Γ⁺-convergence is not topological. Thus convergence in τ_M does not implies Γ-convergence in general. Take a constant net f _λ  : [0, 1] → [0, 1] defined as f λ ( x ) = { ( 1 / 4 , 1 / 4 ) , x > 0 ; ( 0 , 1 ) , x = 0 .

Then it is not Γ⁺-convergent to itself, as (Lsf _λ ) (0) = (1/4, 1/4) and f (0) = (0, 1).

Remark 4.8. All the above definitions and results are also valid if we replace IF ( X ) by USC ( X ) .

Rest of our discussion is dedicated for upper semicontinuous intuitionistic fuzzy sets.

It is already shown that Γ-convergence implies convergence in topological space ( USC ( X ) , τ M ) . The main work of this section is to find, when the converse is true? Here on, it is assumed that (X, τ _X ) is Hausdroff topological space.

Proposition 5.1. Let {f _λ } _λ∈Λ be a net in USC ( X ) . It is convergent to f ∈ USC ( X ) in the topology τ M + if and only if for x in X f ( x ) ≥ inf U ∈ N x sup K ∈ C ( U ) lim sup λ ∈ Λ M K ( f λ ) , where C (U) is the set of all compact subsets of U.

Proof. First for further reference, let ( G f λ ) ( x ) : = inf U ∈ N x sup K ∈ C ( U ) lim sup λ ∈ Λ M K ( f λ ) .(5.1)

Assume that {f _λ } _λ∈Λ is convergent to f in τ M + . By Remark 4.3 M K ( f ) ≥ lim sup λ ∈ Λ M K ( f λ ) , for any compact subset K of X. Hence for U ∈ N x sup K ∈ C ( U ) M K ( f ) ≥ sup K ∈ C ( U ) lim sup λ ∈ Λ M K ( f λ ) .

Thus, inf U ∈ N x sup K ∈ C ( U ) M K ( f ) ≥ ( G f λ ) ( x ) .

Also f is upper semicontinuous, so f ( x ) ≥ inf U ∈ N x M U ( f ) .

Conversely, assume that f (x) ≥ (Gf _λ ) (x). Consider a compact subset K of X. If M _K  (f) = (1, 0) then of course M K ( f ) ≥ lim sup λ ∈ Λ M K ( f λ ) . Otherwise take ζ > M _K  (f) then for all x in K ζ > (Gf _λ ) (x). Hence for any x ∈ K, we can find U x ∈ N x such that lim sup λ ∈ Λ M L ( f λ ) < ζ for any L ∈ C ( U x ) and K ⊆ ⋃ x ∈ K U x . But K is compact, there will be finite elements x₁, x₂, …, x _n in X such that K ⊆ ⋃ i = 1 n U x i .

There is some compact subset L _i of U _{x i} , such that K ⊆ ⋃ i = 1 n L i , so M K ( f λ ) ≤ M ⋃ i = 1 n L i ( f λ ) .

But lim sup λ ∈ Λ M L i ( f λ ) < ζ for all i = 1, 2, …, n. It means lim sup λ ∈ Λ M K ( f λ ) < ζ . As this is true all ζ > M _K  (f), hence M K ( f ) ≥ lim sup λ ∈ Λ M K ( f λ ) . ■

Remark 5.2. A net {f _λ } _λ∈Λ in USC ( X ) is convergent to f ∈ USC ( X ) in topological space ( USC ( X ) , τ M ) if and only if for x ∈ X ( G f λ ) ( x ) ≤ f ( x ) ≤ ( Li f λ ) ( x ) .

Also, as (Gf _λ ) (x) ≤ (Lsf _λ ) (x) therefore if {f _λ } _λ∈Λ is Γ-convergent to f then f ( x ) = ( Li f λ ) ( x ) = ( Ls f λ ) ( x ) for x ∈ X .

Hence, it is again concluded that Γ-convergence implies convergence in τ_M. Converse is not true as the following example shows.

Example 5.3. Take infinite dimensional Hilbert space X endowed with its weak topology. Let { e n } n ∈ ℕ be a sequence of orthonormal elements of X and < . , .> stands for dot product. Consider a sequence of IFSs on X f n ( x ) = { ( | < x , e n > | 2 n , n - < x , e n > 2 n ) , < x , x > < n ; ( 1 , 0 ) , < x , x > ≥ n . Then Li end f n = Ls end f n = X × 𝕋 . To prove this fact it is enough to show (x, (1, 0)) ∈ Liendf _n . Consider the hyperplane X _n  = {y ∈ X : < y, e _n  > = n} then f _n  (y) = (1, 0) for all y ∈ X _n . Now for x ∈ X if every U ∈ N x has residually non empty intersection with { X n } n ∈ ℕ then we are done. Take U ∈ N x , U = { y ∈ X , | < y - x , v i > | < ɛ , i = 1 , 2 , 3 , … , m } and let Y = span (v₁, v₂, …, v _m ).

It is needed to proof that if U ∩ X _m  = ∅ thene _m  ∈ Y .

If e _m  ∉ Y then there is some z ∈ Y^⊥ for which <z, e _m  > ≠0. Take y = x + m - < x , e m > < z , e m > z , then y ∈ U ∩ X _m . Since dimension of Y is finite, so there will be some n ∈ ℕ such that for all m ≥ n U ∩ X _m  ≠ ∅ . Hence, { f n } n ∈ ℕ is Γ-convergent to f where f (x) = (1, 0) for all x ∈ X.

On the other hand any compact subset of X is bounded and also { f n ( x ) } n ∈ ℕ is convergent to (0, 1/2) for all x in X. Therefore (Gf _n ) (x) = (0, 1/2) for all x ∈ X. Finally by Remark 5.2 the net { f n } n ∈ ℕ is convergent to all f ∈ USC ( X ) in the topology τ_M for which (0, 1/2) ≤ f (x) ≤ (1, 0) for all x ∈ X.

For the converse to be true, the topological space (X, τ _X ) needed to be locally compact.

Lemma 5.4. Take a locally compact topological space (X, τ _X ). For a net {f _λ } _λ∈Λ in USC ( X ) ( G f λ ) ( x ) = ( Ls f λ ) ( x ) for all x ∈ X .

Proof. Since (Lsf _λ ) (x) ≥ (Gf _λ ) (x) therefore the reverse inequality is needed to be proved. For x in X if (Gf _λ ) (x) = (1, 0) then of course (Lsf _λ ) (x) = (1, 0), otherwise take ζ ∈ 𝕋 such that (Gf _λ ) (x) < ζ. Thus there will be some U ∈ N x , for which sup K ∈ C ( U ) lim sup λ ∈ Λ M K ( f λ ) < ζ .

But also X is locally compact, let K₀ be the compact neighbourhood of x. As X is Hausdroff there will be V ∈ N x such that V ¯ ⊂ K 0 ∩ U . So V ¯ ∈ C ( U ) , hence ( Ls f λ ) ( x ) ≤ lim sup λ ∈ Λ M V ( f λ ) < ζ .

It is true for all such ζ, thus (Lsf _λ ) (x) ≤ (Gf _λ ) (x) . ■

Theorem 5.5. Consider (X, τ _X ) is locally compact space. Then a net {f _λ } _λ∈Λ in USC ( X ) is

Γ^--convergence to f if and only if it is convergent to f in topological space ( USC ( X ) , τ M - ) ;

Proof. (1): Follows from Proposition 5.1.

(2): Follows from Proposition 5.1 and Lemma 5.4.

(3): Follows from (1) and (2). ■

Example 5.3 shows that ( USC ( X ) , τ M ) is not Hausdroff. In the end, it is proved that Local compactness is also necessary and sufficient for ( USC ( X ) , τ M ) to be Hausdroff.

Theorem 5.6. The topological space ( USC ( X ) , τ M ) is Hausdroff if and only if X is locally compact space.

Proof. Let (X, τ _X ) be locally compact. Take f₁ ≠ f₂ in USC ( X ) then there is x ∈ X such that f₁ (x) ≠ f₂ (x). There are two cases.

Case (1): f₁ (x) and f₂ (x) are compare able, without loss suppose that f₁ (x) < f₂ (x). Take ζ ∈ 𝕋 for which f₁ (x) < ζ < f₂ (x). Since f₁ is upper semicontinuous and X is locally compact we can find a compact neighbourhood K of x such that M _K  (f₁) < ζ. But also f₂ is upper semicontinuous and X is Hausdroff, therefore there is some U ∈ N x such that U ⊆ K and M _U  (f₂) > ζ. Thus f₂ ∈ A_U,ζ, f₁ ∈ B_K,ζ and A_U,ζ∩ B_K,ζ = ∅ because U ⊆ K.

Case (2): f₁ (x) and f₂ (x) are not comparable, without loss assume that α_f2(x) > α_f1(x) and β_f2(x) > β_f1(x). Take ζ 1 , ζ 2 ∈ 𝕋 such that f₁ (x) < ζ₁ < max(f₁ (x), f₂ (x)) and ζ₂ = (α _{ζ 1} , β_f2(x)), as shown in the Fig. 1.

OPEN IN VIEWER

Fig.1

Case (2).

Then by arguments similar to case (1), there is compact subset K of X and U ∈ τ _X subset of K, such that M _K  (f₁) < ζ₁ and M _U  (f₂) > ζ₂. Hence f₁ ∈ B_K,ζ1 and f₂ ∈ A_U,ζ2 and A_U,ζ2∩ B_K,ζ1 = ∅.

Conversely, suppose that ( USC ( X ) , τ M ) is Hausdroff. Fix x in X and consider following IFSs f 1 ( y ) = { ( 1 / 2 , 3 / 10 ) , y = x ; ( 1 / 4 , 1 / 2 ) , otherwise ; and f 2 ( y ) = ( 1 / 4 , 1 / 2 ) for all y ∈ X .

There will be disjoint open sets V₁ and V₂ containing f₁ and f₂, respectively. They will be of the form V 1 = ⋂ i ∈ Δ 1 , 1 A U 1 , i , ζ 1 , i ∩ ⋂ j ∈ Δ 1 , 2 B K 1 , j , ζ 1 , j ; and V 2 = ⋂ i ∈ Δ 2 , 1 A U 2 , i , ζ 2 , i ∩ ⋂ j ∈ Δ 2 , 2 B K 2 , j , ζ 2 , j ; where cardinality of Δ_l,k for l = 1, 2 and k = 1, 2 is finite. Let Δ = { i ∈ Δ 1 , 1 : U 1 , i ∈ N x } . Then ζ_1,i < (1/2, 3/10) for all i ∈ Δ, ζ_1,i < (1/4, 1/2) for all i ∈ Δ_1,1 ∖ Δ, ζ_2,i < (1/4, 1/2) for all i ∈ Δ_2,1 and (1/4, 1/2) < ζ_l,j for all j ∈ Δ_l,2 : l = 1, 2 .

Now denote K = ⋃ j ∈ Δ 1 , 2 K 1 , j ∪ ⋃ j ∈ Δ 2 , 2 K 2 , j , this compact set is the required compact neighbourhood of x. Suppose contrary that U_1,i ⊈ K for all i ∈ Δ, so there is some x _i in U_1,i ∖ K. Let S be the set of all such x _i . Take f ( y ) : = { ( 1 , 0 ) , y ∈ S ; ( 1 / 4 , 1 / 2 ) , otherwise , then f ∈ V₁ ∩ V₂, a contradiction. Hence there is some i ∈ Δ such that U_1,i ⊆ K. ■