Husain [1] introduced the idea of supra-open sets in 1977, which is thought to be an additional category of several recognized varieties of near-open sets. Mashhour et al. [2] established S continuity for a single-valued function f: (ℵ, Ω) → (ℋ, Γ) in 1983. Many topological features of these and other concepts have been established [2–7]. The (upper/lower) supra-continuous multifunction (brevity supra-CMF) is intended to be presented in this study as an extension of each of (upper/lower) semi-CMF in the sense of [8–14], the (upper/lower) quasi-continuous (brevity QC), the (upper/lower) pre-CMF, and weak continuous (brevity WC) for functions due to Popa [15, 16], (upper/lower) α-continuous (brevity α-C) given by Neubrunn [17], (upper/lower) β-CMF defined by Popa and Noiri [18–20]. Furthermore, several features associated with these novel multifunctions have been established.

The remainder of this paper is organized as follows. Our main contribution is to define and explore the features of weak continuous multifunctions, including their applications. This method provides a more complex understanding. The use of supra-continuity allows for a more accurate simulation of uncertainty and imprecision in real-world circumstances. In Section 2, we discuss the fundamentals. Section 3 presents the concept of supra-continuous multifunctions. Section 4 investigates supra-continuous multifunctions and supra-closed graphs. Finally, Section 5 concludes the paper with suggestions for future research.

This section includes a variety of basic concepts and symbols that will be used in the ensuing text. Henceforth, we refer to the multifunction (MF), continuous multifunction (CMF), single-valued function (SV F), regular multifunction (RMF), supra-compact (supra-Com), supra-convergent (supra-Conv), supra-regular (supra-r), paracompact (Pcom), and continuous (C).

Here, the topological spaces denoted by (ℵ, Ω) and (ℋ, Γ) were employed. In contrast, Ω-cl(Z) and Ω-int(Z) are indicated by the closure and interior of any subset Z of ℵ regarding a topology Ω. In (ℵ, Ω), the class Ω^* ⊑ ℘(ℵ) is called a supra-topology on ℵ if ℵ ∈ Ω^* and Ω^* are closed under arbitrary union [1]. (ℵ,Ω^*) is a supra-topological space (brevity supra-space), each member of Ω^* is supra-open, and its complement is supra-closed [2]. Let (ℵ, Ω) be a topological space and Ω^* be a supra-topology on ℵ; Ω^* is associated with Ω if Ω ⊑ Ω^* [2]. Throughout this study, we always consider Ω^* and Γ^* to be associated with Ω and Γ, respectively. In (ℵ,Ω₁), the supra-closure, supra-interior, and supra-frontiers of any ℏ ⊑ ℵ are indicated by supra-cl(ℏ), supra-int(ℏ) and supra-fr(ℏ), respectively, as defined in [2]. For any n ∈ ℵ, Ω^*(n) = {Z ⊑ ℵ: Z ∈ Ω^*, n ∈ Z}. In (ℵ, Ω), ℏ ⊑ ℵ is called semi-open [21]. If one is present U ∈ Ω that U ⊑ ℏ ⊑ Ω-cl(U) and ℏ is preopen [4] if ℏ ⊑ Ω-int(Ω-cl(ℏ)). The families of all semi-open and preopen sets in (ℵ, Ω) are denoted by SO(ℵ, Ω) and PO(ℵ, Ω), respectively. Moreover, Ω^α = SO(ℵ, Ω) ⊓ PO(ℵ, Ω) and SO(ℵ, Ω) ⊔ PO(ℵ, Ω) ⊑ βO(ℵ, Ω). However, ℏ ∈ Ω^α and ℏ ∈ βO(ℵ, Ω) are known as α-set [22] and β-open set [23], respectively. An SV F f: (ℵ, Ω) → (ℋ, Γ) is known as S-C [2] if each V ∈ Γ, f⁻¹(V) ∈ Ω^* where Ω^* is associated with Ω. For a multifunction f: (ℵ, Ω) → (ℋ, Γ), the upper and lower inverse of any N ⊑ ℵ₁ will be given by f⁺(N) = {n ∈ ℵ: f(n) ⊑ N} and f⁻ (N) = {n ∈ ℵ: f(n) ⊓ N ≠ φ}, respectively. In addition, f: (ℵ, Ω) → (ℋ, Γ) is known as the upper (resp. lower) semi-C [8], if for each V ∈ Γ, f⁺(Z) ∈ ℵ (resp.f⁻ (Z) ∈ ℵ). If Ω in semi-continuity is replaced by SO(ℵ, Ω),Ω^α, PO(ℵ, Ω) and βo(ℵ, Ω), then f is upper/lower quasi-C [15], upper/lower α-C [17], upper/lower pre-C [16], and upper/lower β-C [18]. A supra-space (ℵ,Ω^*) is called supra-Com [5], if every supra-open cover of ℵ admits a finite subcover. A subset Z of a space (ℵ, Ω) is called α-precompact [24] if for every open cover Ψ of Z in (ℵ, Ω) there exists a locally finite open cover ζ of Z that refines Ψ. An MF f: (ℵ, Ω) → (ℋ, Γ) is punctually α-PCom if for each n ∈ ℵ, f(n) is α-Pcom.

In this section, we focus on providing the supra-CMF, punctually α-Pcom, and punctually supra-RMF.

Definition 3.1

An MF f: (ℵ, Ω) → (ℋ, Γ) is said to be:

• 1) upper supra-C at point n ∈ ℵ if every f(n)-containing open set V of ℋ contains Z ∈ Ω^* such that f(Z) ⊑ V where Ω^* is associated with Ω.

• 2) lower supra-C at a point n ∈ ℵ if every f(n)-containing open set V of ℋ, there is Z ∈ Ω^* that f(Z) ⊓ V ≠ φ.

• 3) f is upper(lower) supra-C if this feature exists at all points of ℵ.

Any SVF f: (ℵ, Ω) → (ℋ, Γ) is a multi-valued function allocated to any n ∈ ℵ, the singleton {f(n)} using SV by applying the definitions of the upper and lower supra-CMF given above. These clearly correspond to Mashhour et al.’s [2] definition of S-C.

Remark 3.2

For an MF f: (ℵ, Ω) → (ℋ, Γ), many properties of the upper/lower semi-continuity [8] (resp. upper/lower ℵ-C [17], upper/lower quasi-C [15], upper/lower pre-C [16], and upper/lower β-C [18])) can be deduced from the upper/lower supra-C by considering Ω^* = Ω (resp. Ω^* = Ω^α,Ω^* = SO(ℵ, Ω),Ω^* = PO(ℵ, Ω) and Ω^* = βO(ℵ, Ω).

One characterization of the aforementioned MF is established throughout the following conclusion, which has a simple proof, thus it is removed.

Proposition 3.3

An MF f: (ℵ, Ω) → (ℋ, Γ) is the upper/lower supra-C at point n ∈ ℵ if and only if for any Z ∈ Γ with f(n) ⊑ Z/f(n) ⊓ Z ≠ φ. Then n ∈ supra-int(f⁺(Z))/n ∈ supra-int(f⁻(Z)).

Lemma 3.4

For any ℏ ⊑ (ℵ, Ω),Ω-int(ℏ) ⊑supra-int(ℏ) ⊑ supra-cl(ℏ) ⊑ Ω-int(ℏ).

Theorem 3.5

For MF f: (ℵ, Ω) → (ℋ, Γ), the following are identical:

• 1) f is the upper supra-C.

• 2) f⁺(V) ∈ Ω^*, for each and every V ∈ Γ.

• 3) f⁻(K) is supra-closed for each and every closed set K ⊑ ℋ.

• 4) supra-cl(f⁻(N)) ⊑ f⁻(ℋ-cl(N)), for each and every N ⊑ ℋ.

• 5) f⁺(Ω₁-int(N)) ⊑ supra-int(f⁺(N)), for each and every N ⊑ ℋ.

• 6) supra-fr(f⁻(N)) ⊑ f⁻(fr(N)), for each and every N ⊑ ℋ.

• 7) f: (ℵ, Ω) → (ℋ, Γ) is upper semi-C.

Proof. 1) ⇔ 2): Following from Proposition 3.3.

2) ⇒ 3): Let K be closed in ℋ, the result is satisfied because f⁺(ℋ \ K = ℵ \ f⁻(K).

3) ⇒ 4): By placing K = Γ-cl(N) and making use of Lemma 3.4.

4) ⇒ 5): Let N ⊑ ℋ, then Γ-int(N) ∈ Γ; hence, ℋ \ Γ-int(N) is closed in (ℋ, Γ).

Therefore through 4) we obtain ℵ\supra-int(f⁺(N)) =supra-cl(ℵ\ f⁺(N)) ⊑supra-cl(ℵ\f⁺(ℋ-int(N)) =supra-cl(f⁻(ℋ\ Γ-int(N)) ⊑ f⁻(ℋ \ Γ-int(N)) ⊑ ℵ \ f⁺(Γ-int(N)). This implies that f⁺(Ω-int(N)) ⊑ supra-int(f⁺(N)).

5) ⇒ 1): Let n ∈ ℵ be arbitrary and each V ∈ Ω(f(n)) then f⁺(V) ⊑ supra-int(f⁺(V)). This indicates that f⁺(V) ∈ Ω^*(n), and 1) will be verified by considering Z = f⁺(V).

4) ⇒ 3): Clearly, supra-frontier and frontier of any set is supra-closed and closed, respectively.

7) ⇒ 2): Occurs immediately.

Theorem 3.6

For MF f: (ℵ, Ω) → (ℋ, Γ), the ensuing claims are comparable:

• 1) f is lower than supra-C.

• 2) f⁻(V) ∈ Ω^*, for every V ∈ Γ.

• 3) f⁺(K) is supra-closed for every closed set K ⊑ ℋ.

• 4) supra-cl(f⁺(N)) ⊑ f⁺(Ω₁-cl(N)), for every N ⊑ ℋ.

• 5) f⁻(Γ-int(N)) ⊑ supra-int(f⁻(N)), for every N ⊑ ℋ.

• 6) supra-fr(f⁺(N)) ⊑ f⁺(fr(N)), for every N ⊑ ℋ.

• 7) f: (ℵ, Ω) → (ℋ, Γ) is lower semi-C.

Proof. This proof strongly resembles that of Theorem 3.5.

Remembering that the net (n_i)_i∈I is supra-Conv to n_◦, if each Z ∈ ℵ^*(n_◦) there is an i_◦ ∈ I such that each i > i_◦ implies n_i ∈ Z.

Theorem 3.7

An MF f: (ℵ, Ω) → (ℋ, Γ), is upper supra-C if and only if for each net supra-Conv to n_◦ and each V ∈ Γ with f(n_◦) ⊑ V ∃ i_◦ ∈ I such that f(n_i) ⊑ V for all i > i_◦.

Proof. Let V ∈ Γ with f(n_◦) ⊑ V. From the upper supra-C of f, there exists Z ∈ Ω^*(n_◦) such that f(Z) ⊑ V. From the hypothesis, a net (n_i)_i∈I is supra-Conv to n_◦ and Z ∈ Ω^*(n_◦) there is i_◦ ∈ I such that n_i ∈ Z for all i ≥ i_◦ and f(n_i) ⊑ V for all i ≥ i_◦.

Being self-sufficient, suppose the opposite, that is, there is an open set V in ℋ with f(n_◦) ⊑ V such that for each Z ∈ Ω^*, having f(Z) ⋢ V, that is, there is n_Z ∈ Z where f(n_Z) ⋢ V. Under the inclusion relation, all n_Z form a net in ℵ with a directed set Z of Ω^*(n_◦); this net is supra-Conv to n_◦. However, f(n_Z) ⋢ V for all Z ∈ Ω^*(n_◦). This results in a contradiction that concludes the proof.

Theorem 3.8

An MF f: (ℵ, Ω) → (ℋ, Γ), is lower supra-C if and only if each h_◦ ∈ f(n_◦) and for every net (n_i)_i∈I supra-Conv to n_◦, there exists a subset (w_j)_j∈J of the net (n_i)_i∈I and a net (h_j)_(j,V)∈J in ℋ such that (h_j)_(j,V)∈J supra-Conv to h and h_j ∈ f(w_j).

Proof. Suppose f is lower than supra-C, (n_i)_i∈I is a net supra-Conv to n_◦, h ∈ f(n_◦) and V ∈ Γ(h). Thus, we have f(n_◦)⊓ V ≠ φ because of the lower supra-C of f at n_◦, there exists a supra-open set Z ⊑ ℵ containing n_◦ such that Z ⊑ f⁻(V). Supra-Conv of a net (n_i)_i∈I to n_◦ and for this Z, there is an i_◦ ∈ I such that for each i ≥ i_◦, n_i ∈ Z and therefore n_i ∈ f⁻(V). Hence, for each V ∈ Γ(h), define the sets I_v = {i_◦ ∈ I: i ≥ i_◦ ⇒ n_i ∈ f⁻(V)} and J = {(i, V): V ∈ d(h), i ∈ I_v} and order ≥ on J given as (í, V́) ≥ (i, V) if and only if í ≥ i and V́ ⊑ V. In addition, define ξ: J → I as ξ((j́, V́)) = j. Then, ξ increases and is cofinal in I; therefore, ξ defines a subset of (n_i)_i∈I denoted by (w_j)_(j,V)∈J. However, for any (j, V) ∈ J because j ≥ j_◦ implies n_i ∈ f⁻(V), f(w_j) ⊓ V = f(n_j)⊓V ≠ φ. Select h_j ∈ f(w_j)⊓V ≠ φ. Subsequently, net (h_j)_(j,V)∈J is supra-Conv to h. To see this, let V_◦ ∈ Γ(h), then there is j_◦ ∈ I with j_◦ = ξ((j_◦, V_◦)); (j_◦, V_◦) ∈ J and h_j◦ ∈ V. If (j, V) ≥ (j_◦, V_◦), then j ≥ j_◦ and V ⊑ V_◦. Therefore, h_j ∈ f(w_j) ⊓ V ⊑ f(n_j) ⊓ V ⊑ f(n_j) ⊓ V_◦ and h_j ∈ V_◦. Thus, (h_j)_(j,V)∈J is supra-Conv to h, which demonstrates this result. To show sufficiency, assume the converse; that is, f is not lower supra-C at n_◦. Then, there exists V ∈ Γ such that f(n_◦) ⊓ V ≠ φ and for any supra-neighborhood Z ⊑ ℵ of n_◦, there is n_z ∈ Z for which f(n_z) ⊓ V ≠ φ. Let us consider net (n_z)_{z∈Ω^*(n◦)} which is obviously supra-Conv to n_◦. Suppose h_◦ inf(n_◦) ⊓ V ≠ φ by hypothesis, there is a subnet (w_k)_k∈K of (n_Z)_{Z∈Ω^*(n◦)} and h_k ∈ f(w_k) like (h_k)_k∈K supra-Conv to h_◦. Because h_◦ ∈ V ∈ Γ, ḱ_◦ ∈ K exists such that k ≥ ḱ_◦ implies h_k ∈ V. By contrast (w_k)_k∈K is a subnet of the net (n_z)_{z∈Ω^*(n◦)} and there is a function Ψ: K → Ω^*(n_◦) such that w_k = n_Ω(K) and for each Z ∈ Ω^*(n_◦) there exists k´´o∈K such that Ω(k´´o)≥Z. If K≥k´´o then Ω(k)≥Ω(k´´o)≥Z. Consider k_◦ ∈ K such that k_◦ ≥ ḱ_◦ and Ko≥k´´o. Therefore, h_k ∈ V and the meaning of the net (n_z)_{z∈Ω^*(n◦)}, we have f(w_k) ⊓ V = f(n_Ω(k)) ⊓ V = φ. This results in h_k ∉ V, which defines the hypothesis and satisfies the criteria.

Definition 3.9

A subset Z of a space (ℵ, Ω) is named supra-r, if for any n ∈ Z and H ∈ Ω^*(n) there is U ∈ Γ such that n ∈ U ⊑ Γ-cl(U) ⊑ H. Therefore, recall that f: (ℵ, Ω) → (ℋ, Γ) is punctually supra-r if for each n ∈ ℵ, f(n) is supra-r.

Lemma 3.10

Given that Z ⊑ ℵ is supra-r and is contained in a supra-open set H, there exists a space (ℵ, Ω). Next, U ∈ Ω exists, such that Z ⊑ U ⊑ Γ-cl(U) ⊑ H.

For an MF f: (ℵ, Ω) → (ℋ, Γ), MF supra-cl(f): (ℵ, Ω) → (ℋ, Γ) is defined as: (supra-clf)(n) =supra-cl(f(n)) for every n ∈ ℵ.

Proposition 3.11

For a punctually α-paracompact and punctually supra-rMF f: (ℵ, Ω) → (ℋ, Γ). Then, (supra-cl(f))⁺ (Z)) = f⁺(Z) for each Z ∈ Γ^*.

Proof. Consider n ∈(supra-cl(f))⁺(Z)) for any Z ∈ Γ^*, implying that f(n) ⊑ supra-cl(f(n)) ⊑ Z which results in n ∈ f⁺(Z). Therefore, only one inclusion is considered valid. To demonstrate this, let n ∈ f⁺(Z) where Z ∈ Γ^*(n). Then, f(n) ⊑ Z based on the hypothesis of f and the fact that Γ ⊑ Γ^* with Lemma 3.10, exists G ∈ Γ such that f(n) ⊑ G ∈ Γ-cl(G) ⊑ Z. Therefore, supra-cl(f(n)) ⊑ Z implies n ∈(supra-cl(f))⁺(Z)). Thus, the equality is verified.

Theorem 3.12

Let f: (ℵ, Ω) → (ℋ, Γ) be punctually α-PCom and supra-rMF. Then, f is the upper supra-C if and only if (supra-clf): (ℵ, Ω) → (ℋ, Γ) is the upper supra-C.

Proof. We consider V ∈ Γ and n ∈(supra-cl(f))⁺(V) = f⁺(V) (see Proposition 3.11). The upper supra-C of f implies that there exists H ∈ Ω^*(n) such that f(H) ⊑ V. As Γ ⊑ Γ^*, from Lemma 3.11 and the assumption that f, there exists G ∈ Γ that f(h) ⊑ G ⊑ Γ-cl(G) ⊑ Z ∀h ∈ H. Hence, supra-cl(f(h)) =(supra-clf)(h) ⊑ supra-cl(G) ⊑ Γ-cl(G) ⊑ V for each h ∈ H, yielding (supra-clf)(H) ⊑ V. Thus, (supra-clf) is the upper-C.

Sufficiency, suppose V ∈ Γ and n ∈ f⁺(V) =(supra-clf)⁺(V). According to the hypothesis of f in this case, H ∈ Ω^*(n) such that (supra-clf)(H) ⊑ V which gives f(H) ⊑ V. The proof is complete.

Lemma 3.13

In space (ℵ, Ω), any n ∈ ℵ and ℏ ⊑ ℵ, n ∈ supra-cl(ℏ) if and only if ℏ ⊓ Z ≠ φ for each Z ∈ Ω^*(n).

Proposition 3.14

For MF f: (ℵ, Ω) → (ℋ, Γ), (supra-cl(f))⁻(Z)) = f⁻(Z) for every Z ∈ Γ^*.

Proof. Let n ∈(supra-cl(f))⁻ (Z)); then Z⊓supra-cl(f(n))) ≠ φ. As Z ⊑ Γ^*, Lemma 3.13 yields Z ⊓ f(n) ≠ φ; hence, n ∈ f⁻(Z). Conversely, let n ∈ f⁻(Z) then, φ ≠ f(n) ⊓ Z ⊑(supra-clf)⁻(n) ⊓ Z; thus, n ∈(supra-clf)⁻(n) ⊓ Z, and the equality is completed.

Theorem 3.15

MF f: (ℵ, Ω) → (ℋ, Γ) is a lower supra-C if and only if (supra-clf): (ℵ, Ω) → (ℋ, Γ) is a lower supra-C.

Proof. This result is directly derived from Proposition 3.11 considering that Ω ⊑ Ω^* and 2) of Theorem 3.2.

Theorem 3.16

If f: (ℵ, Ω) → (ℋ, Γ) is an upper supra-C surjection and for each n ∈ ℵ, f(n) is compact relative to ℋ. If (ℵ, Ω) is supra-Com, then (ℋ, Γ) is compact.

Proof. Let {V_i: i ∈ I, V_i ∈ Γ} be the cover of ℋ, and f(n) is compact relative to ℋ for each n ∈ ℵ. Then there exists a finite I_◦(n) of I such that f(n) ⊑ ⊔{V_i: i ∈ I_◦(n)}. Given the upper supra-C of f, Z(n) ∈ Ω^*(ℵ, n) exists such that f(Z(n)) ⊑ ⊔{V_i: i ∈ I_◦(n)}. Because (ℵ, Ω) is supra-Com, then there exists {n₁, n₂, ..., n_m} such that ℵ = ⊔{Z(n_j): 1 ≤ j ≤ m}. Therefore, ℋ = f(ℵ) = ⊔{f(Z(n_j)): 1 ≤ j ≤ m} ⊑ ⊔{V_i: i ∈ I_◦(n_j), 1 ≤ j ≤ m}. Therefore, (ℋ, Γ) is compact.

In this section, we present supra-CMF and supra-closed graphs related to α-Pcom.

Definition 4.1

If there exist Z ∈ Ω^*(n) and H ∉ Γ^*(h) such that (Z × H) ⊓ G(f) = φ for each pair (n, h) ∉ G(f), then MF f: (ℵ, Ω) → (ℋ, Γ) has a supra-closed graph. A multifunction f: (ℵ, Ω) → (ℋ, Γ) is point-closed (supra-closed) if for each n ∈ ℵ, f(n) is closed (supra-closed) in ℋ.

Proposition 4.2

MF f: (ℵ, Ω) → (ℋ, Γ) has a supra-closed graph if and only if there are two supra-open sets H, Z containing n and h, respectively, such that f(H) ⊓ Z = φ for each n ∈ ℵ and h ∈ ℋ such that h ∉ f(n).

Proof. Necessity, let n ∈ ℵ and h ∈ ℋ with h ∉ f(n). Then, using a supra-closed graph of f, there exist H ∈ ℵ^*(n) and Z ∈ Γ^* containing f(n) such that (H × Z) ⊓ G(f) = φ. This suggests that f(H) ⊓ Z = φ for n ∈ H and h ∈ Z where h ∉ f(n).

Sufficiency, let (n, h) ∉ G(f), this means h ∉ f(n) then f(H) ⊓ Z = φ for each of the two disjoint supra-open sets H, Z that contain n, h. The proof is completed by implying that (H × Z) ⊓ G(f) = φ.

Theorem 4.3

If f: (ℵ, Ω) → (ℋ, Γ) is the upper supra-C and point-closed MF. Subsequently, G(f) is supra-closed if (ℋ, Γ) is regular.

Proof. Assume that h ∉ f(n) if (n, h) ∉ G(f). Because ℋ is regular, there exists a disjoint V_i ∈ Γ, i = 1, 2 such that h ∈ V₁ and f(n) ⊑ V₂. As f is the upper supra-C at n, there exists Z ∈ Ω^*(n) such that f(Z) ⊑ V₂. As V₁ ⊓ V₂ = φ, then ⊓i=12 supra-int(V_i) ∉ φ and therefore, n ∈supra-int(Z) = Z, h ∈supra-int(V₁) and (n, h) ∈ Z×supra-int(V₁) ⊑ (ℵ × Γ) \ G(f). Thus, (ℵ × Γ) \ G(f) ∈ ℵ^*(ℵ × Γ) yields the result.

Theorem 4.4

Let f: (ℵ, Ω) → (ℋ, Γ) be an upper supra-CMF from (ℵ, Ω) to the Hausdorff space (ℋ, Γ). If f(n) is α-PCom for each n ∈ ℵ, G(f) is supra-closed.

Proof. Assume that h_◦ ∉ f(n_◦) and (n_◦, h_◦) ∉ G(f). Since (ℋ, Γ) is Hausdorff, then for each h_◦ ∈ f(n_◦) there exists V_h ∈ Γ(h) and Vh*∈Γ(ho) such that Vh⊓Vh*=φ. Thus, the family {V_h: h ∈ f(n_◦)} is the open cover of f(n_◦). Thus, from α-PCom of f(n_◦), v = {U_i: i ∈ I} is a locally finite open cover that refines {V_h: h ∈ f(n_◦)}. Consequently, H_◦ ∈ Γ(h_◦) exists, where H_◦ intersects only a finite number of members, U_i1, U_i2, ..., U_im of v. Choose h₁, h₂, ..., h_m in f(n_◦) such that U_ij ⊑ V_hj for each 1 ≤ j ≤ m, and the set H = H_◦ ⊓ (⊔_i∈IV_hj). Then, H ∈ Γ(h_◦) such that H ⊓ (⊔_i∈IV_i) = φ. Given the upper supra-C of f, n_◦ ∈ Z ⊑ f⁺(⊔_i∈IV_i) exists for any Z ∈ Ω^*(n_◦). Consequently, G(f) is supra-closed because (Z ⊓ H) ⊓ G(f) = φ.

Lemma 4.5 [25]

The subsequent hold for f: (ℵ, Ω) → (ℋ, Γ), ℏ ⊑ ℵ and N ⊑ ℋ;

• 1) Gf+(ℏ×N)=ℏ⊓f+(N);

• 2) Gf-(ℏ×N)=ℏ⊓f-(N),

where G(f) means the graph MF of f given by G_f (ℵ, Ω) → (ℵ×ℋ,Ω×Γ) and defined by G_f (n) = {n}×f(n), for every n ∈ ℵ.

Theorem 4.6

For an MF f: (ℵ, Ω) → (ℋ, Γ), if G_f is the upper supra-C, then f is the upper supra-C.

Proof. Let n ∈ ℵ and V ∈ Γ(f(n)). As ℵ × V ∈ Ω × Γ and G_f (n) ⊑ ℵ × V, according to Theorem 3.5, Z ∈ ℵ^*(n) exists such that G_f (Z) ⊑ ℵ × V. Consequently, f(Z) ⊑ V is obtained using Lemma 4.5 to Z⊑Gf-(ℵ×V)=ℵ⊓Gf+(V)=f+(V). Thus, from Theorem 3.5 f upper supra-C.

Theorem 4.7

If the graph G_f of an MF f: (ℵ, Ω) → (ℋ, Γ) is lower supra-C, then f is also so.

Proof. Let n ∈ ℵ and V ∈ Γ(f(n)) with f(n) ⊓ V ≠ φ, also As ℵ×V ∈ Ω×Γ then G_f (n)⊓(ℵ×V) = {{n}×f(n)}⊓(ℵ× V) = {n}×(f(n)⊓V) ≠ φ. From Theorem 3.6, for any z ∈ Z, there exists Z ∈ ℵ^*(n) such that G_f (Z) ⊑ ℵ × V ≠ φ. Hence from Lemma 4.5, Z⊑Gf-(ℵ×V)=ℵ⊓Gf-(V)=f-(V). Thus, the proof is complete for each z ∈ Z using f(Z)⊓V ≠ φ and Theorem 3.6.

In future work, we will investigate novel WC values for MF, CMF and other properties. In addition, in the future it can be studied Pythagorean fuzzy soft somewhat CMF. We introduced new notions for continuous multifunctions, where we made a generalization of the WC of MFs. Several characterizations and fundamental features of these new types of MF were obtained, as well as the relationship between supra-CMF and supra-closed graphs.