EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6155
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Multifunctions Between Temporal Picture Fuzzy Ideal
Structures

Dali Shi1, M.N. Abu_Shugair2,∗, S.E. Abbas3, Ismail Ibedou4

1 Gugangzhou College of Technology and Business, China
2 Mathematics Department, College of Science, Jazan University, Jazan 45142, Saudi Arabia
3 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
4 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Abstract. This paper joins the notion of multifunctions to the notion of a temporal picture fuzzy
modal topological structures (TPFMTS) using ideals. In this paper, we introduce the notion
of temporal picture fuzzy local function and TPF-ideal topological spaces. Also, we introduce
the concepts of TPFu or TPF l LP -continuous, almost LP -continuous, weakly LP -continuous
and almost weakly LP -continuous multifunctions. Several properties and characterizations of the
presented multifunctions and their types of continuity are established. Some examples are given
to explain the correct implications between these notions.
2020 Mathematics Subject Classifications: AMS 94D05, 03E72, 03E75, 03B52, 03B20
Key Words and Phrases: Temporal Picture Fuzzy Multifunction, Temporal Picture Fuzzy Set,
Temporal Picture Fuzzy Topological Structure, Picture Fuzzy Ideal

1. Introduction

Fuzzification is a crucial tool for addressing humanistic systems in our real-life problems.
The first paper on fuzzy set theory was authored by Zadeh in 1965 ([1]). This approach of
fuzzy sets FS has been widely applied by many scholars. Fuzzy sets described the positivism
of an element ϱ of a universal set ℵ to a subset G ⊆ ℵ by the membership value ωG(ϱ), and
posited that the negativism of that element ϱ ∈ ℵ to the set G is 1− ωG(ϱ). Atanassov in
[2] based his theory of intuitionistic fuzzy sets IFS on the notion of the negativism ϖG(ϱ)
of an element ϱ ∈ ℵ to a subset G ⊆ ℵ that may range from [0, 1] and need not be the
complement of the positivism of that element ϱ ∈ ℵ to G. The values ωG(ϱ) and ϖG(ϱ)
represent the positivism and negativism of each ϱ ∈ ℵ to G, respectively, with the condition
that 0 ≤ ωG(ϱ) + ϖG(ϱ) ≤ 1. In this way, Atanassov encompassed all the fuzzy sets FS
as a special case of his theory whenever ωG(ϱ) +ϖG(ϱ) = 1. Intuitionistic fuzzy sets IFS

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6155

Email addresses: shidali@gzgs.edu.cn (D. Shi), mabushqair@jazanu.edu.sa (M. N. Abu_Shugair),
salaheldin_ahmed@science.sohag.edu.eg (S. E. Abbas), ismail.abdelaziz@fsc.bu.edu.eg (I. Ibedou)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 2 of 25

are more meaningful and applicable to our real-life cases. Cuong in [3] initiated the theory
of picture fuzzy sets PFS by adding the neutralism of an element ϱ ∈ ℵ to the subset G,
represented by σG(ϱ). This definition is conditioned with 0 ≤ ωG(ϱ) +ϖG(ϱ) + σG(ϱ) ≤ 1.
In case where σG(ϱ) = 0 for all ϱ ∈ ℵ, then we revert to intuitionistic sets G in IFS.
Moreover, if ϖG(ϱ) = 1− ωG(ϱ) for all ϱ ∈ ℵ, then we revert to a fuzzy set G in FS. There
are several simple modifications for the intuitionistic fuzzy sets [2], which we shall not
discuss here. These modifications include Pythagorean fuzzy sets [4, 5], Fermatean fuzzy
sets [6], Spherical fuzzy sets [7], q-rung orthopair fuzzy sets [8–10] and q-rung orthopair
picture fuzzy sets [11]. Moreover, (ς, κ)-fuzzy local function, continuous multifunctions
and DF-ideal topological space are found in [12–14]. All these definitions, starting from
fuzzy sets, have applications in image processing, decision theory, uncertainty modeling,
and beyond, as in [5–7, 15–17].

In this paper, we compile the definitions and notions from regions of general topology,
of standard modal logic [16, 18–21] and of picture fuzziness, and represent the concept of a
temporal picture fuzzy modal topological structure, briefly TPFMTS. Continuous functions
between picture fuzzy topological spaces were discussed in [22]. Some types of continuity
of multifunctions were studied in [23, 24]. Also, we presented the general form of temporal
picture fuzzy continuous multifunctions. In this paper, we merge the classical definitions
of multifunctions in general topology and the standard modal logic [16, 19–21, 25] with
the notion of picture fuzzy sets, further expanding into the realm of TPFMTS.

The motivations of this paper are as follow. Section 1 is an introduction. Section 2
is given for the basics. Section 3 presents the main definition of temporal picture fuzzy
local functions that are joined to a temporal picture fuzzy ideal. Section 4 investigates
the notions of temporal picture upper and temporal picture lower almost -LP -continuity
and introduces many characteristic properties of these defined multifunctions. Section 5
investigates the notions of temporal picture upper and temporal picture lower weak -LP -
continuity and discusses its properties, as well as investigates the implications associated
with the previous definitions of temporal picture upper and temporal picture lower almost-
LP -continuity. Section 6 investigates the notions of temporal picture upper and temporal
picture lower almost weak -LP -continuity, and discusses its properties, as well as inves-
tigates the implications associated with the previous definitions. Section 7 presents the
conclusion.

The research on TPFMTS has several important applications in various domains: De-
cision Making, Pattern Recognition, Artificial Intelligence, Information Retrieval and Data
Mining. TPFMTS address critical gaps in handling uncertainty, imprecision, and neutral-
ity, which are inherent in real-life problems across diverse domains. To bridge these gaps,
Picture Fuzzy Sets PFS were introduced, adding a neutrality component to the mem-
bership and non-membership values, thereby enabling a more nuanced representation of
uncertainty. TPFMTS expands upon these concepts by the integration in modal logic and
general topology using the picture fuzzy sets. This integration introduces global operators,
such as closure and interior, which modify classical topological and modal relationships.
These global operators facilitate a robust analysis of fuzzy sets under modal and topological
constraints, providing a suitable tools for theoretical exploration and practical application.



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 3 of 25

The study of TPFMTS not only extends the theory of fuzzy sets but also establishes a wide
platform for addressing modern computational challenges. Its ability to integrate neutral-
ity, positivity, and negativity within a unified framework lays the foundation for further
exploration and application of TPFMTS in dynamic systems, hybrid models, and emerging
technologies, positioning it as a cornerstone of modern mathematical and computational
innovation.

2. Preliminaries

Through the paper, denote I = [0, 1], I0 = (0, 1] and I1 = [0, 1). Let a universe set ℵ, be
fixed. An PFS G in ℵ is an object of the following form: G = {⟨ϱ, ωG(ϱ), ϖG(ϱ), σG(ϱ)⟩ |ϱ ∈
ℵ}, where ωG(ϱ) ∈ I is called the degree of positive membership of ϱ in G, ϖG(ϱ) ∈ I is
called the degree of negative membership of ϱ in G, σG(ϱ) ∈ I is called the degree of neutral
membership of ϱ in G, and where ωG(ϱ), ϖG(ϱ) and σG(ϱ) satisfy the following condition:
0 ≤ ωG(ϱ) +ϖG(ϱ) + σG(ϱ) ≤ 1 for all ϱ ∈ ℵ. Let G be a non-empty set (finite or infinite),
called the temporal scale with upper and lower boundaries. The elements of G are called
“time-moments”. we define the temporal PFS (TPFS) as follows:

G (G) = {⟨⟨ϱ, g⟩ , ωG(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩), σG(⟨ϱ, g⟩)⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G}, where
(1) G ⊆ ℵ is a fixed set,
(2)ωG(⟨ϱ, g⟩) +ϖG(⟨ϱ, g⟩) + σG(⟨ϱ, g⟩) ≤ 1 for every ⟨ϱ, g⟩ ∈ ℵ ×G,
(3)ωG(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩) and σG(⟨ϱ, g⟩) are the degree of positive membership, negative

membership and neutral membership, respectively, of the element ϱ ∈ ℵ at the time-
moment g ∈ G.

NOTE: each ordinary PFS can be regarded as a TPFS for which G is a singleton set.
Additionally, we mentioned that all operations and operators on the PFSs can be

defined for the TPFSs. However, the opposite is also true: each TPFS G (G)is a standard
PFS, but over universe ℵ × G. For this reason, we can re-define all operations, relations,
and operators defined over standard PFSs, now over TPFSs.

Definition 2.1. [22, 26] Let ℵ be a nonempty set, G time-scale, G (G) = {⟨⟨ϱ, g⟩ , ωG(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩), σG(⟨ϱ, g⟩)⟩
| ⟨ϱ, g⟩ ∈ ℵ ×G} and H (G) = {⟨⟨ϱ, g⟩ , ωH(⟨ϱ, g⟩), ϖH(⟨ϱ, g⟩), σH(⟨ϱ, g⟩)⟩ | ⟨ϱ, g⟩
∈ ℵ ×G}. Then,
(1) G (G) ⊆ H (G)iff ∀ ⟨ϱ, g⟩ ∈ ℵ ×G, ωG(⟨ϱ, g⟩) ≤ ωH(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩) ≥ ϖH(⟨ϱ, g⟩)
and σG(⟨ϱ, g⟩) ≤ σH(⟨ϱ, g⟩) or σG(⟨ϱ, g⟩) ≥ σH(⟨ϱ, g⟩).
(2)G (G)∪H (G) ={⟨⟨ϱ, g⟩ , ωG(⟨ϱ, g⟩) ∨ ωH(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩) ∧ϖH(⟨ϱ, g⟩), σG(⟨ϱ, g⟩) ∧ σH(⟨ϱ, g⟩)⟩ |
⟨ϱ, g⟩ ∈ ℵ ×G}}.
(3)G (G)∩H (G) ={⟨⟨ϱ, g⟩ , ωG(⟨ϱ, g⟩) ∧ ωH(⟨ϱ, g⟩), ϖG(⟨ϱ, g⟩) ∨ϖH(⟨ϱ, g⟩), σG(⟨ϱ, g⟩) ∧ σH(⟨ϱ, g⟩)⟩ |
⟨ϱ, g⟩ ∈ ℵ ×G}}.
(4) Ⅎ G (G) = {⟨⟨ϱ, g⟩ , ϖG(⟨ϱ, g⟩), ωG(⟨ϱg⟩), σG(⟨ϱ, g⟩)⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G}.
(5) G (G) ⊼H (G) = {⟨⟨ϱ, g⟩ , 0, 1, 0⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G} if G (G) ⊆ H (G) , and

G (G)⊼ H (G) = G (G) ∩ (Ⅎ H (G)) otherwise.
(6) ♯ (G) = {⟨⟨ϱ, g⟩ , 1, 0, 0⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G}, ♭ (G) = {⟨⟨ϱ, g⟩ , 0, 1, 0⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G}.
(7) ♮ (G) = {⟨⟨ϱ, g⟩ , 0, 0, 1⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G},0 (G) = {⟨⟨ϱ, g⟩ , 0, 0, 0⟩ | ⟨ϱ, g⟩ ∈ ℵ ×G}.



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 4 of 25

The map F : ℵ×G ↬ Υ×G is called a temporal picture fuzzy multifunction (TPFM ,
for short) for any ⟨ϱ, g⟩ ∈ ℵ×G, F(⟨ϱ, g⟩) ∈

(
I3
)Υ×G. The degree of membership of ⟨ζ, g⟩ ∈

Υ×G at the time-moment g ∈ G is denoted by: F(⟨ϱ, g⟩)(⟨ζ, g⟩) = ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩). The
domain of F, denoted by D (F) and the range of F, denoted by R (F), are defined by: for
any ⟨ϱ, g⟩ ∈ ℵ × G and ⟨ζ, g⟩ ∈ (Υ×G), D (F) (⟨ϱ, g⟩) =

⋃
⟨ζ,g⟩∈(Υ×G)

ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) and

R (F) (⟨ζ, g⟩) =
⋃

⟨ϱ,g⟩∈ℵ×G

ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩). F is called crisp iff ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) = ♯ (G) ∀

⟨ϱ, g⟩ ∈ ℵ × G and ⟨ζ, g⟩ ∈ (Υ×G). F is called Normalized (NTPFM , for short) iff ∀
⟨ϱ, g⟩ ∈ ℵ×G, there exists ⟨ζ0, g⟩ ∈ (Υ×G) such that ΨF(⟨ϱ, g⟩ , ⟨ζ0, g⟩) = ♯ (G). F is called
surjective iff R (F) ⟨ζ, g⟩ = ♯ (G) ∀ ⟨ζ, g⟩ ∈ (Υ×G). The inverse of F denoted by F− :
Υ ↬ ℵ is a TPFM defined by: F−(⟨ζ, g⟩)(⟨ϱ, g⟩) = F(⟨ϱ, g⟩)((⟨ζ, g⟩)) = ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩).
One easily verifies that D (F−) =R (F) and D (F) = R(F−). The image FG (G)) of G (G) ∈(
I3
)ℵ×G, the lower inverse Fl(U (G)) and the upper inverse Fu(U (G)) of U (G) ∈

(
I3
)Υ×G

are defined respectively (see [8, 27]) as follows:

F(G (G)) ⟨ζ, g⟩ =
⋃

⟨ϱ,g⟩∈ℵ×G

[ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ∩G (G) (⟨ϱ, g⟩)] ,

Fl(U (G)) ⟨ϱ, g⟩ =
⋃

⟨ζ,g⟩∈(Υ×G)

[ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ∩ U (G) ⟨ζ, g⟩] ,

Fu(U (G)) ⟨ϱ, g⟩ =
⋂

⟨ζ,g⟩∈(Υ×G)

[Ⅎ ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ∪ U (G) ⟨ζ, g⟩] .

Definition 2.2. [12] A temporal picture fuzzy topology on ℵ is a map τ :
(
I3
)ℵ×G → I3

defined by
τ(G (G)) = ⟨ωτ (G (G)), ϖτ (G (G)), στ (G (G))⟩ which satisfies the following properties:

(1) τ(♭ (G)) = τ(♯ (G)) = ⟨1, 0, 0⟩.
(2) τ(G ∩H) ≥ τ(G) ∧ τ(H), for each G,H ∈

(
I3
)ℵ×G

.

(3) τ(
⋃
i∈Γ

Gi) ≥
∧
i∈Γ

τ(Gi), for each Gi ∈
(
I3
)ℵ×G, i ∈ Γ.

The pair (ℵ, τ) is called a temporal picture fuzzy topological space in Šostak’s sense.
For any G (G) ∈

(
I3
)ℵ×G the number ωτ (G (G)) is called the openness degree at a certain

times , ϖτ (G (G)) is called the non openness degree at a certain times, while στ (G (G)) is
called the neutral degree at a certain times. For G (G) ∈

(
I3
)ℵ×G

clτ (G (G) , ⟨ς,κ, ϑ⟩) =
⋂

{H (G) ∈
(
I3
)ℵ×G

: G (G) ⊆ H (G) , τ(Ⅎ H (G)) ≥ ⟨ς,κ, ϑ⟩},

intτ (G (G) , ⟨ς,κ, ϑ⟩) =
⋃

{H (G) ∈
(
I3
)ℵ×G

: G (G) ⊇ H (G) , τ(H (G)) ≥ ⟨ς,κ, ϑ⟩}.

Definition 2.3. [12] Let F : (ℵ, τ) ↬ (Υ, σ) be a TPFM , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.
Then, F is called:



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 5 of 25

(1) TPF uS-continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G))

for each U (G) ∈
(
I3
)Υ×G

, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) , such that G (G) ∩D (F) ⊆ Fu(U (G)).

(2) TPF lS-continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G))

for each U (G) ∈
(
I3
)Υ×G

, σU (G)) ≥ ⟨ς,κ, ϑ⟩, there exists G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ , and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) , such that G (G) ⊆ Fl(U (G)).

Remark 2.1. (1) If F is NTPFM , then F is TPF uS -continuous at a fuzzy point
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G)) for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists

G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ , and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) , such that G (G) ⊆
Fu(U (G)).

Definition 2.4. [13] The map LP :
(
I3
)ℵ×G → I3 is called temporal picture fuzzy ideal

on ℵ if it satisfies the following conditions for G (G) , H (G) ∈
(
I3
)ℵ×G:

(1) LP (♭ (G)) = ⟨1, 0, 0⟩ ,LP (♯ (G)) = ⟨0, 1, 0⟩.
(2) G (G) ⊆ H (G) ⇒ LP (G (G)) ≥ LP (H (G)).
(3) LP (G (G) ∪H (G)) ≥ LP (G (G)) ∧ LP (H (G)).

Also, LP is called proper if LP (♯ (G)) = ⟨0, 1, 0⟩ and there exists G (G) ∈
(
I3
)ℵ×G such

that
LP (G (G)) > ⟨0, 1, 0⟩.

If LP
1 and LP

2 are temporal picture fuzzy ideals on ℵ × G, we say that LP
1 is finer

than LP
2 (LP

2 is coarser than LP
1 ), denoted by LP

2 ⊆ LP
1 , iff LP

2 (G (G)) ≤ LP
1 (G (G))

∀G (G) ∈
(
I3
)ℵ×G. Let us define the special picture fuzzy ideals LP0, LP1 by

LP0 (G (G)) =

{
⟨1, 0, 0⟩ if G (G) = ♭ (G) ,
⟨0, 1, 0⟩ otherwise, and LP1 (G (G)) =

{
⟨0, 1, 0⟩ if G (G) = ♯ (G) ,
⟨1, 0, 0⟩ otherwise.

3. Temporal picture fuzzy local functions

Definition 3.1. Let (ℵ, τ,LP ) be a temporal picture fuzzy ideal topological space,
G (G) ∈

(
I3
)ℵ×G, ς ∈ I0, κ ∈ I1 and ϑ ∈ I1. Then the ⟨ς,κ, ϑ⟩-temporal fuzzy local

function Φ(G (G) , ⟨ς,κ, ϑ⟩) of G (G) defined as follows: Φ(G (G) , ⟨ς,κ, ϑ⟩) =
⋂
{H (G) ∈(

I3
)ℵ×G:LP (G (G) ⊼H (G)) ≥ ⟨ς,κ, ϑ⟩,τ(Ⅎ H (G)) ≥ ⟨ς,κ, ϑ⟩}.

Remark 3.1. (1) If we take LP = LP0 for each G (G) ∈
(
I3
)ℵ×G we have

Φ(G (G) , ⟨ς,κ, ϑ⟩) =
⋂
{H (G) ∈

(
I3
)ℵ×G

: G (G) ⊆ H (G), τ(Ⅎ H (G)) ≥ ⟨ς,κ, ϑ⟩} =
clτ (G (G) , ⟨ς,κ, ϑ⟩).

(2) If we take LP = LP1 (resp. LP (G (G)) ≥ ⟨ς,κ, ϑ⟩) for each G (G) ∈
(
I3
)ℵ×G we

have
Φ(G (G) , ⟨ς,κ, ϑ⟩) = ♭ (G).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 6 of 25

We will occasionally write Φ(G (G) , ⟨ς,κ, ϑ⟩) or Φ(G (G) ,LP , ⟨ς,κ, ϑ⟩) for Φ(G (G) ,LP , τ, ⟨ς,κ, ϑ⟩).

Theorem 3.1. Let (ℵ, τ,LP ) be a temporal picture fuzzy ideal topological space and LP
1 ,LP

2

be two temporal picture fuzzy ideals on ℵ. Then for any set G (G) ,H (G) ∈
(
I3
)ℵ×G

, ς ∈
I0,κ ∈ I1 and ϑ ∈ I1.

(1)Φ(♭ (G) , ⟨ς,κ, ϑ⟩) = ♭ (G) .
(2) If G (G) ⊆ H (G)then Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(H (G) , ⟨ς,κ, ϑ⟩).
(3) If LP

2 ⊆ LP
1 then Φ(G (G) ,LP

1 , ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G) ,LP
2 , ⟨ς,κ, ϑ⟩).

(4)Φ(G (G) , ⟨ς,κ, ϑ⟩) = cl (Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩) .
(5)Φ(Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G) , ⟨ς,κ, ϑ⟩) and Ⅎ (Φ(G (G) , ⟨ς,κ, ϑ⟩)) ̸=

Φ(Ⅎ G (G) , ⟨ς,κ, ϑ⟩).
(6)Φ(G (G)∪ H (G) , ⟨ς,κ, ϑ⟩) ⊇ Φ(G (G) , ⟨ς,κ, ϑ⟩)∪Φ(H (G) , ⟨ς,κ, ϑ⟩) and Φ(G (G)∩

H (G) , ⟨ς,κ, ϑ⟩)
⊆ Φ(G (G) , ⟨ς,κ, ϑ⟩) ∩ Φ(H (G) , ⟨ς,κ, ϑ⟩).
(7) If LP (H (G)) ≥ ⟨ς,κ, ϑ⟩ then Φ(G (G) ∪H (G) , ⟨ς,κ, ϑ⟩) ⊇ Φ(G (G) , ⟨ς,κ, ϑ⟩).

Proof. (1) From Definition 3.1, we have Φ(♭ (G) , ⟨ς,κ, ϑ⟩) = ♭ (G).
(2) Suppose that Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊈ Φ(H (G) , ⟨ς,κ, ϑ⟩) If G (G) ⊆ H (G). By the

definition of
Φ(H (G) , ⟨ς,κ, ϑ⟩),there exists K (G) ∈

(
I3
)ℵ×G with Φ(H (G) , ⟨ς,κ, ϑ⟩) ⊆ K (G), LP (H (G)⊼

K (G)) ≥ ⟨ς,κ, ϑ⟩, τ(Ⅎ K (G)) ≥ ⟨ς,κ, ϑ⟩. Such that Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊈ K (G). Since
G (G) ⊆ H (G)implise G (G) ⊼ K (G) ⊆ H (G) ⊼ K (G), LP (G (G) ⊼ K (G)) ≥ LP (H (G) ⊼
K (G)) ≥ ⟨ς,κ, ϑ⟩. Hence Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆ K (G), it is a contradiction. Then,
Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(H (G) , ⟨ς,κ, ϑ⟩).

(3) Suppose that Φ(G (G) ,LP
1 , ⟨ς,κ, ϑ⟩) ⊈ Φ(G (G) ,LP

2 , ⟨ς,κ, ϑ⟩) if LP
2 ⊆ LP

1 . By the
definition of
Φ(G (G) ,LP

2 , ⟨ς,κ, ϑ⟩) there exists K (G) ∈
(
I3
)ℵ×G with Φ(G (G) ,LP

2 , ⟨ς,κ, ϑ⟩) ⊆ K (G),
LP
2 (G (G) ⊼ K (G)) ≥ ⟨ς,κ, ϑ⟩, τ(Ⅎ K (G)) ≥ ⟨ς,κ, ϑ⟩ such that Φ(G (G) ,LP

1 , ⟨ς,κ, ϑ⟩) ⊈
K (G). Since LP

2 ⊆ LP
1 implies LP

1 (G (G)⊼K (G)) ≥ LP
2 (G (G)⊼K (G)) ≥ ⟨ς,κ, ϑ⟩. Hence,

Φ(G (G) ,LP
1 , ⟨ς,κ, ϑ⟩) ⊆ K (G), it is a contradiction. Then, Φ(G (G) ,LP

1 , ⟨ς,κ, ϑ⟩) ⊆
Φ(G (G) ,LP

2 , ⟨ς,κ, ϑ⟩).
(4) From Definition 3.1, we have Φ(G (G) , ⟨ς,κ, ϑ⟩) = cl (Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩).
Since LP0 ⊆ LP for any picture fuzzy ideal LP , Φ(G (G) ,LP , ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G) ,

LP0, ⟨ς,κ, ϑ⟩) = cl (G (G) , ⟨ς,κ, ϑ⟩). Thus,
Φ(G (G) , ⟨ς,κ, ϑ⟩) = cl (Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩).

(5) By (4), we have Φ(Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = cl (Φ(Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)
⊆ cl (Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = Φ(G (G) , ⟨ς,κ, ϑ⟩). In general the converse is not true
as shown in next Example 3.1.

(6) Since G (G) ⊆ G (G)∪H (G) and H (G) ⊆ G (G)∪H (G)implies Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆
Φ(G (G) ∪H (G) , ⟨ς,κ, ϑ⟩) and Φ(H (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G) ∪H (G) , ⟨ς,κ, ϑ⟩). Thus,
Φ(G (G) , ⟨ς,κ, ϑ⟩)∪Φ(H (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G)∪H (G) , ⟨ς,κ, ϑ⟩). Also, since G (G)∩
H (G) ⊆ G (G)and G (G)∩H (G) ⊆ H (G) implies Φ(G (G)∩H (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(G (G) , ⟨ς,κ, ϑ⟩)
and Φ(G (G)∩H (G) , ⟨ς,κ, ϑ⟩) ⊆ Φ(H (G) , ⟨ς,κ, ϑ⟩). Thus, Φ(G (G)∩H (G) , ⟨ς,κ, ϑ⟩) ⊆
Φ(G (G) , ⟨ς,κ, ϑ⟩) ∩ Φ(H (G) , ⟨ς,κ, ϑ⟩).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 7 of 25

(7) Since LP (H (G)) ⊇ ⟨ς,κ, ϑ⟩ implies Φ(H (G) , ⟨ς,κ, ϑ⟩) = ⟨0, 1, 0⟩ .
Thus,Φ(G (G)∪H (G) , ⟨ς,κ, ϑ⟩) ⊇ Φ(G (G) , ⟨ς,κ, ϑ⟩)∪Φ(H (G) , ⟨ς,κ, ϑ⟩) ⊇ Φ(G (G) , ⟨ς,κ, ϑ⟩).

The following example shows that generally Φ(Φ(G (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ̸= Φ(G (G) , ⟨ς,κ, ϑ⟩)
and
Ⅎ (Φ(G (G) , ⟨ς,κ, ϑ⟩)) ̸= Φ(Ⅎ G (G) , ⟨ς,κ, ϑ⟩) for any G (G) ∈

(
I3
)ℵ×G

, ς ∈ I0,κ ∈
I1 and ϑ ∈ I1.

Example 3.1. Let ℵ = {ϱ1, ϱ2}, G = {g1, g2}, Define τ, LP :
(
I3
)ℵ×G → I3 as follows:

τ(G (G)) =



⟨1, 0, 0⟩ if G (G) ∈ {♭ (G) , ♯ (G)} ,

⟨0.4, 0.3, 0.2⟩ if G (G) = G1 (G) ,

⟨0.5, 0.2, 0.3⟩ if G (G) = G2 (G) ,

⟨0, 1, 0⟩ o.w,

LP (G (G)) =


⟨1, 0, 0⟩ if G (G) = ♭ (G) ,

⟨0.7, 0.2, 0.1⟩ if ♭ (G) ⊂ G (G) ⊆ ⟨⟨ϱ, g⟩ , 0.4, 0.3, 0.3⟩ , ⟨ϱ, g⟩ ∈ ℵ ×G},

⟨0, 1, 0⟩ o.w.

where

G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.3, 0.2⟩ , ϱ ∈ ℵ
⟨⟨ϱ, g2⟩ , 0.35, 0.4, 0.25⟩ , ϱ ∈ ℵ

}
, G2 (G) =

{
⟨⟨ϱ, g1⟩ , 0.55, 0.25, 0.2⟩ , ϱ ∈ ℵ
⟨⟨ϱ, g2⟩ , 0.4, 0.35, 0.25⟩ , ϱ ∈ ℵ

}
,

G3 (G) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.3, 0⟩ , ϱ ∈ ℵ
⟨⟨ϱ, g2⟩ , 0.4, 0.35, 0⟩ , ϱ ∈ ℵ

}
and H (G) =

{
⟨⟨ϱ, g1⟩ , 0.45, 0.35, 0.2⟩ , ϱ ∈ ℵ
⟨⟨ϱ, g2⟩ , 0.44, 0.25, 0.2⟩ , ϱ ∈ ℵ

}
.

Then, ♭ (G) = Φ(Φ(H (G) , ⟨0.4, 0.3, 0.2⟩), ⟨0.4, 0.3, 0.2⟩) ̸= Φ(H (G) , ⟨0.4, 0.3, 0.2⟩)

=

{
⟨⟨ϱ, g1⟩ , 0.3, 0.3, 0⟩ , ϱ ∈ ℵ
⟨⟨ϱ, g2⟩ , 0.4, 0.35, 0⟩ , ϱ ∈ ℵ

}
,

♯ (G) = Ⅎ (Φ(G3 (G) , ⟨0.4, 0.3, 0.2⟩)) ̸= Φ(Ⅎ G3 (G) , ⟨0.4, 0.3, 0.2⟩) = ♭ (G) .

Definition 3.2. Let (ℵ, τ,LP ) be a temporal picture fuzzy ideal topological space. Then,
for each G (G) ∈

(
I3
)ℵ×G, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, we define an operator cl∗ :

(
I3
)ℵ×G×

I3 →
(
I3
)ℵ×G as follows:

cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) = G (G) ∪ Φ(G (G) , ⟨ς,κ, ϑ⟩).

Now, if LP = LP0 then
cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) = G (G) ∪ Φ(G (G) , ⟨ς,κ, ϑ⟩) = G (G) ∪ clτ (G (G) , ⟨ς,κ, ϑ⟩) =

clτ (G (G) , ⟨ς,κ, ϑ⟩) .

Theorem 3.2. Let (ℵ, τ,LP ) be a temporal picture fuzzy ideal topological space. Then
for any fuzzy set G (G) ,H (G) ∈

(
I3
)ℵ×G

, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the operator
cl∗τ :

(
I3
)ℵ×G × I3 →

(
I3
)ℵ×G satisfies the following properties:



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 8 of 25

(1) cl∗τ (♭ (G) , ⟨ς,κ, ϑ⟩) = ♭ (G) .
(2)G (G) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩) .
(3) If G (G) ⊆ H (G), then cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (H (G) , ⟨ς,κ, ϑ⟩).
(4) cl∗τ (G (G) ∪H (G) , ⟨ς,κ, ϑ⟩) ⊇ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ∪ cl∗τ (H (G) , ⟨ς,κ, ϑ⟩).
(5) cl∗τ (G (G) ∩H (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ∩ cl∗τ (H (G) , ⟨ς,κ, ϑ⟩).
Proof. (1) Since cl∗τ (♭ (G) , ⟨ς,κ, ϑ⟩) = ♭ ∪ Φ(♭ (G) , ⟨ς,κ, ϑ⟩) and Φ(♭ (G) , ⟨ς,κ, ϑ⟩) =

♭ (G) implies
cl∗τ (♭ (G) , ⟨ς,κ, ϑ⟩) = ♭ (G) .

(2) cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) = G (G)∪Φ(G (G) , ⟨ς,κ, ϑ⟩) implies G (G) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩),
Since G (G) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩) and from Theorem 3.1(4), we have Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆
cl (G (G) , ⟨ς,κ, ϑ⟩) implies
cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩). Thus, G (G) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ cl (G (G) , ⟨ς,κ, ϑ⟩).

(3) From G (G) ⊆ H (G) and Theorem 3.1(2), we have G (G) ∪ Φ(G (G) , ⟨ς,κ, ϑ⟩) ⊆
H (G) ∪ Φ(H (G) , ⟨ς,κ, ϑ⟩), i.e., cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (H (G) , ⟨ς,κ, ϑ⟩).

(4) Since G (G) ⊆ G (G)∪H (G) and H (G) ⊆ G (G)∪H (G) implies cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆
cl∗τ (G (G)∪H (G) , ⟨ς,κ, ϑ⟩) and cl∗τ (H (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (G (G)∪H (G) , ⟨ς,κ, ϑ⟩). Thus,
cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) ∪ cl∗τ (H (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (G (G) ∪H (G) , ⟨ς,κ, ϑ⟩).

(5) G (G) ∩H (G) ⊆ G (G) and G (G) ∩H (G) ⊆ H (G) implies
cl∗τ (G (G) ∩ H (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩) and cl∗τ (G (G) ∩ H (G) , ⟨ς,κ, ϑ⟩) ⊆
cl∗τ (H (G) , ⟨ς,κ, ϑ⟩). Thus, cl∗τ (G (G)∩H (G) , ⟨ς,κ, ϑ⟩) ⊆ cl∗τ (G (G) , ⟨ς,κ, ϑ⟩)∩cl∗τ (H (G) , ⟨ς,κ, ϑ⟩).

Theorem 3.3. Let (ℵ, τ,LP ) be a picture fuzzy ideal topological space. Then, for each
G (G) ∈

(
I3
)ℵ×G,

ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, we define an operator int∗τ :
(
I3
)ℵ×G× I3 →

(
I3
)ℵ×G as follows:

int∗τ (G (G) , ⟨ς,κ, ϑ⟩) = G (G) ∩ Ⅎ (Φ(Ⅎ G (G) , ⟨ς,κ, ϑ⟩)) .

For G (G), H (G) ∈
(
I3
)ℵ×G, the operator int∗ satisfies the following properties:

(1) int∗τ (♯ (G) , ⟨ς,κ, ϑ⟩) = ♯ (G) .
(2) int (G (G) , ⟨ς,κ, ϑ⟩) ⊆ int∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ G (G) .
(3) If G (G) ⊆ H (G) , then int∗τ (G (G) , ⟨ς,κ, ϑ⟩) ⊆ int∗τ (H (G) , ⟨ς,κ, ϑ⟩).
(4) int∗τ (G (G) ∩H (G) , ⟨ς,κ, ϑ⟩) ⊆ int∗τ (G (G) , ⟨ς,κ, ϑ⟩) ∩ int∗τ (H (G) , ⟨ς,κ, ϑ⟩).
(5) int∗τ (♯, ⟨ς,κ, ϑ⟩) = int (G (G) , ⟨ς,κ, ϑ⟩) if LP = LP0.
(6) int∗τ (Ⅎ G (G) , ⟨ς,κ, ϑ⟩) = Ⅎ (cl∗τ (G (G) , ⟨ς,κ, ϑ⟩)) .

Proof. It is similarly proved as the proof of Theorem 3.2.

Definition 3.3. Let F : (ℵ, τ,LP ) ↬ (Υ, σ) be a TPFM , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.
Then, F is called:

(1) TPF u LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈
Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ∩D (F) ⊆ Φ(Fu(U (G)), ⟨ς,κ, ϑ⟩).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 9 of 25

(2) TPF l LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G))

for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ⊆ Φ(Fl(U (G)), ⟨ς,κ, ϑ⟩).

(3) TPF u LP -continuous (resp. TPF l LP -continuous) iff it is TPF u LP -continuous
(resp. TPF l LP -continuous) at every fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F).

Remark 3.2. (1) If F is NTPFM , then F isTPF u LP -continuous at a fuzzy point
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G)) for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists

G (G) ∈
(
I3
)ℵ×G, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ⊆

Φ(Fu(U (G)), ⟨ς,κ, ϑ⟩).
(2) TPF u (resp. TPF l) LP -continuity and TPF u (resp. TPF lS)-continuity are inde-

pendent notions as shown by Example 3.2.

Theorem 3.4. Let F : (ℵ, τ,LP ) ↬ (Υ, σ) be a TPFM (resp. NTPFM), Then F is
TPF l (resp. TPF u) LP -continuous iff Fl (U (G)) ⊆ intτ

(
Φ(Fl(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩

)
(resp. Fu(U (G) ⊆ intτ (Φ(F

u(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)) for each U (G)∈
(
I3
)Υ×G

, σ(U (G)) ≥
⟨ς,κ, ϑ⟩ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1 .

Proof. (⇒) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then, there exists G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that
G (G) ⊆ Φ(Fl(U (G)), ⟨ς,κ, ϑ⟩). Thus, ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) ⊆ intτ (Φ(F

l(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩),
and hence

Fl(U (G)) ⊆ intτ (Φ(F
l(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩).

(⇐) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈

Fl(U (G)). Then, F l (U (G)) ⊆ intτ (Φ(F
l(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) and hence,

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ intτ (Φ(F
l(U (G)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Φ(Fl(U (G)), ⟨ς,κ, ϑ⟩).

Thus, F is TPF l LP -continuous. Other case is similarly proved.

The following examples show that there is no relation between TPF u (TPF l) S-
continuous and TPF u (TPF l) LP -continuous multifunctions.

Example 3.2. Let ℵ = {ϱ1, ϱ2}, Υ = {ζ1, ζ2, }, G = {g1, g2} and F : ℵ ↬ Υ be a TPFM
defined by
ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) as:

ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ⟨ζ1, g1⟩ ⟨ζ1, g2⟩ ⟨ζ2, g1⟩ ⟨ζ2, g2⟩
⟨ϱ1, g1⟩ ⟨0.2, 0.15, 0.3⟩ ⟨0.15, 0.3, 0.5⟩ ⟨1, 0, 0⟩ ⟨0, 1, 0⟩
⟨ϱ1, g2⟩ ⟨1, 0, 0⟩ ⟨0.1, 0.15, 0.4⟩ ⟨0.3, 0.25, 0.1⟩ ⟨0.3, 0.2, 0.3⟩
⟨ϱ2, g1⟩ ⟨1, 0, 0⟩ ⟨0.3, 0.3, 0.4⟩ ⟨0.2, 0.2, 0.2⟩ ⟨0.3, 0.25, 0.2⟩
⟨ϱ2, g2⟩ ⟨0.5, 0.15, 0.15⟩ ⟨0.25, 0.2, 0.1⟩ ⟨1, 0, 0⟩ ⟨0.2, 0.1, 0.2⟩ .



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 10 of 25

Define temporal picture fuzzy topologies τ1, τ2:
(
I3
)ℵ×G → I3, σ1, σ2:

(
I3
)Υ×G → I3,

and temporal picture fuzzy ideals LP
1 , LP

2 :
(
I3
)Υ×G → I3as:

τ1(G (G)) =


⟨1, 0, 0⟩ , G (G) ∈ {♭ (G) , ♯ (G)}

⟨0.6, 0.3, 0.1⟩ , G (G) = G1 (G)

⟨0, 1, 0⟩ , o.w,

τ2(G (G)) =


⟨1, 0, 0⟩ , G (G) ∈ {♭ (G) , ♯ (G)}

⟨0.4, 0.15, 0.35⟩ , G (G) = G2 (G)

⟨0, 1, 0⟩ , o.w,

LP
1 (G (G)) =



⟨1, 0, 0⟩ , G (G) = ♭ (G)

⟨0.4, 0.2, 0.3⟩ , ♭ (G) ⊂ G (G) ⊆ ⟨⟨ϱ, g⟩ , 0.4, 0.1, 0.4⟩ ,
⟨ϱ, g⟩ ∈ ℵ ×G

⟨0, 1, 0⟩ , o.w,

LP
2 (G (G)) =



⟨1, 0, 0⟩ , G (G) = ♭ (G)

⟨0.5, 0.15, 0.3⟩ , ♭ (G) ⊂ G (G) ⊆ ⟨⟨ϱ, g⟩ , 0.25, 0.25, 0.5⟩ ,
⟨ϱ, g⟩ ∈ ℵ ×G

⟨0, 1, 0⟩ , o.w,

σ(U(G)) =


⟨1, 0, 0⟩ , U (G) ∈ {♭ (G) , ♯ (G)}

⟨0.3, 0.3, 0.4⟩ , U(G) = U1(G)

⟨0, 1, 0⟩ , o.w.

where

G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0.2⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0.3⟩ ,

, ϱ ∈ ℵ
}

, G2 (G) =

{
⟨⟨ϱ, g1⟩ , 0.5, 0.4, 0.1⟩ ,
⟨⟨ϱ, g2⟩ , 0.6, 0.2, 0.1⟩ ,

, ϱ ∈ ℵ
}

and U1 (G) =

{
⟨⟨ζ, g1⟩ , 0.3, 0.33, 0.35⟩ ,
⟨⟨ζ, g2⟩ , 0.33, 0.33, 0.33⟩ ,

, ζ ∈ Υ

}
.

Then, (1) F : (ℵ, τ1, ℓP1 ) ↬ (Υ, σ) is TPF uS (resp. TPF lS)-continuous but it is not
TPF u (resp. TPF l) ℓP -continuous because

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ (F
u(U1 (G), ⟨0.3, 0.3, 0.4⟩)

=

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}
,

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ (F
l(U1 (G), ⟨0.3, 0.3, 0.4⟩)



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 11 of 25

=

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}
,

but

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ (Φ(F
u(U1 (G)), ⟨0.3, 0.3, 0.4⟩), ⟨0.3, 0.3, 0.4⟩)

= ♭ (G),

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ (Φ(F
l(U1 (G)), ⟨0.3, 0.3, 0.4⟩), ⟨0.3, 0.3, 0.4⟩)

= ♭ (G).
(2) F : (ℵ, τ2, ℓP2 ) ↬ (Υ, σ) is TPF u (resp. TPF l) ℓP -continuous but it is not TPF uS

(resp.TPF lS)-continuous, because,

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ (Φ(F
u(U1 (G)), ⟨0.3, 0.3, 0.4⟩), ⟨0.3, 0.3, 0.4⟩)

= ♯ (G) ,

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ (Φ(F
l(U1 (G)), ⟨0.3, 0.3, 0.4⟩), ⟨0.3, 0.3, 0.4⟩)

= ♯ (G) ,
but

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ (F
u(U1 (G), ⟨0.3, 0.3, 0.4⟩) = ♭ (G) ,

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.33, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.3, 0.33, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ (F
l(U1 (G), ⟨0.3, 0.3, 0.4⟩) = ♭ (G)

.

4. Temporal picture fuzzy almost continuous multifunctions

Definition 4.1. Let F : (ℵ, τ,LP ) ↬ (Υ, σ) be a TPFM , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.
Then, F is called:

(1) TPF uA LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈
Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G)

such that G (G) ∩D (F) ⊆ Fu(intσ(cl
∗
σ(U (G) , ⟨ς,κ, ϑ⟩ , ⟨ς,κ, ϑ⟩)).

(2) TPF lA LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈
Fl(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G such that G (G) ⊆ Fl(intσ(cl

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)).

(3) TPF uA LP -continuous (resp. TPF lA LP -continuous) iff it is TPF uA LP -continuous
(resp. TPF lA LP -continuous) at every fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F).

Remark 4.1. (1) If F is NTPFM , then F is TPF uA LP -continuous at a fuzzy point
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥

⟨ς,κ, ϑ⟩ there exists
G (G) ∈

(
I3
)ℵ×G, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that

G (G) ⊆ Fu(intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 12 of 25

(2) TPF uS (resp. TPF lS)-continuity ⇒ TPF uA (resp. TPF lA) LP -continuity ⇒
TPF uA (resp. TPF lA) -continuity.

(3) TPF uA (resp. TPF lA) LP0-continuity ⇔ TPF uA (resp. TPF lA) -continuity.

Theorem 4.1. For a TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF lA LP -continuous.
(2) Fl(U (G)) ⊆ intτ

(
Fl(intσ(cl

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)

)
, ⟨ς,κ, ϑ⟩),if σ(U (G)) ≥

⟨ς,κ, ϑ⟩ .
(3) clτ (F

u(clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu (U (G)), if σ(Ⅎ U (G)) ≥

⟨ς,κ, ϑ⟩ .

Proof. (1) =⇒ (2) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then, there exists G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that
G (G) ⊆ Fl(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)).

Thus, ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) ⊆ intτF
l(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), and

hence
Fl (U (G)) ⊆ intτ

(
Fl(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩

)
.

(2) =⇒ (3) Let U (G) ∈
(
I3
)Υ×G with σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩. Then by (2) ,

Ⅎ Fu (U (G)) = Fl(Ⅎ U (G)) ⊆ intτ

(
Fl(intσ(cl

∗
σ (Ⅎ U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩

)
= Ⅎ clτ (F

u(clσ(int
∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) .

Thus clτ (F
u(clσ(int

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu (U (G)).

(3) =⇒ (1) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then by (3), we have

Ⅎ
[
intτ

(
Fl(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)

)
, ⟨ς,κ, ϑ⟩)

]
= clτ (F

u(clσ(int
∗
σ (Ⅎ U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu (Ⅎ U (G)) = Ⅎ Fl(U (G)),

and Fl(U (G)) ⊆ intτ
(
Fl(intσ(cl

∗
σ (K (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)

)
, ⟨ς,κ, ϑ⟩). Therefore,

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ intτ
(
Fl(intσ(cl

∗
σ (K (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)

)
, ⟨ς,κ, ϑ⟩)

⊆ Fl(intσ(cl
∗
σ (K (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). Thus, F is TPF lA LP -continuous.

The following theorem is similarly proved as the proof of Theorem 4.1.

Theorem 4.2. For a NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF uA LP -continuous.
(2) Fu (U (G)) ⊆ intτ (F

u(intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) , ⟨ς,κ, ϑ⟩), if σ(U (G)) ≥

⟨ς,κ, ϑ⟩ .
(3) clτ

(
Fl(clσ(int

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩

)
⊆ Fl (U (G)), if σ(Ⅎ U (G)) ≥

⟨ς,κ, ϑ⟩ .



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 13 of 25

The following examples show that the inverse implications in Remark 4.1(2) are not
satisfied.

Example 4.1. Let ℵ = {ϱ1, ϱ2}, Υ = {ζ1, ζ2, }, G = {g1, g2} and F : ℵ ↬ Υ be a TPFM
defined by
ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) as :

ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ⟨ζ1, g1⟩ ⟨ζ1, g2⟩ ⟨ζ2, g1⟩ ⟨ζ2, g2⟩
⟨ϱ1, g1⟩ ⟨0, 1, 0⟩ ⟨0.2, 0.3, 0.5⟩ ⟨1, 0, 0⟩ ⟨0.25, 0.25, 0.5⟩
⟨ϱ1, g2⟩ ⟨0.3, 0.15, 0.55⟩ ⟨0.25, 0.3, 0.4⟩ ⟨1, 0, 0⟩ ⟨0.3, 0.5, 0.2⟩
⟨ϱ2, g1⟩ ⟨1, 0, 0⟩ ⟨0.3, 0.3, 0.4⟩ ⟨0.25, 0.5, 0.25⟩ ⟨0.15, 0.15, 0.15⟩
⟨ϱ2, g2⟩ ⟨1, 0, 0⟩ ⟨0.4, 0.1, 0.3⟩ ⟨0.2, 0.3, 0.1⟩ ⟨0.4, 0.2, 0.4⟩ .

Define temporal picture fuzzy topologies τ :
(
I3
)ℵ×G → I3, σ:

(
I3
)Υ×G → I3 , and

temporal picture fuzzy ideal LP :
(
I3
)Υ×G → I3 as:

τ(G (G)) =


⟨1, 0, 0⟩ , G (G) ∈ {♭ (G) , ♯ (G)}

⟨0.6, 0.1, 0.3⟩ , G (G) = G1 (G)

⟨0, 1, 0⟩ , o.w,

σ(U(G)) =


⟨1, 0, 0⟩ , U (G) ∈ {♭ (G) , ♯ (G)}

⟨0.31, 0.31, 0.38⟩ , U(G) = U1(G)

⟨0, 1, 0⟩ , o.w.

LP (U(G)) =



⟨1, 0, 0⟩ , U(G) = ♭ (G)

⟨0.4, 0.25, 0.35⟩ , ♭ (G) ⊂ U(G) ⊆ ⟨⟨ζ, g⟩ , 0.3, 0.2, 0.1⟩ ,
⟨ζ, g⟩ ∈ Υ×G

⟨0, 1, 0⟩ , o.w,

where

G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.6, 0.2, 0.1⟩ ,
⟨⟨ϱ, g2⟩ , 0.5, 0.3, 0.2⟩ ,

, ϱ ∈ ℵ
}

, G2 (G) =

{
⟨⟨ϱ, g1⟩ , 0.4, 0.4, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.4, 0.4, 0⟩ ,

, ϱ ∈ ℵ
}

and U1 (G) =

{
⟨⟨ζ, g1⟩ , 0.4, 0.4, 0.2⟩ ,

⟨⟨ζ, g2⟩ , 0.44, 0.41, 0.15⟩ ,
, ζ ∈ Υ

}
.

Then, F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uA (resp. TPF lA) LP -continuous but is not
TPF uS
(resp. TPF lS)-continuous, because Fu(U1 (G)) = G2 (G)

⊆ intτ (F
u(intσ(cl

∗
σ (U1 (G) , ⟨0.31, 0.31, 0.38⟩) , ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) =

♯ (G) ,
Fl(U1 (G)) = G2 (G)
⊆ intτ

(
Fl(intσ(cl

∗
σ (U1 (G) , ⟨0.31, 0.31, 0.38⟩) , ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) =

♯ (G) ,



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 14 of 25

but
Fu(U1 (G)) = G2 (G)⊈ intτ (F

u(U1(G))), ⟨0.31, 0.31, 0.38⟩) = ♭ (G) ,
Fl(U1 (G)) = G2 (G)⊈ intτ

(
Fl(U1(G))), ⟨0.31, 0.31, 0.38⟩

)
= ♭ (G) .

Example 4.2. From the Example 4.1, define temporal picture fuzzy ideal LP :
(
I3
)Υ×G →

I3 as follows:

LP (U(G)) =



⟨1, 0, 0⟩ , U(G) = ♭ (G)

⟨0.55, 0.25, 0.2⟩ , ♭ (G) ⊂ U(G) ⊆
{

⟨⟨ζ, g1⟩ , 0.4, 0.4, 0.1⟩ ,
⟨⟨ζ, g2⟩ , 0.44, 0.41, 0.1⟩ ,

, ζ ∈ Υ

}
⟨0, 1, 0⟩ , o.w,

Then, F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uA (resp. TPF lA)-continuous but is not TPF uA
(resp. TPF lA) LP -continuous because

Fu(U1 (G)) = G2 (G)
⊆ intτ (F

u(intσ(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) = ♯ (G) ,
Fl(U1 (G)) = G2 (G)
⊆ intτ

(
Fl(intσ(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) = ♯ (G) ,

but
Fu(U1 (G)) = G2 (G)
⊈ intτ (F

u(intσ(cl
∗
σ(U1 (G), ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) = ♭ (G) ,

Fl(U1 (G)) = G2 (G)
⊈ intτ

(
Fl(intσ(cl

∗
σ(U1 (G), ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) = ♭ (G) .

Theorem 4.3. For a TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF lA LP -continuous.
(2) τ

(
Fl (U (G))

)
≥ ⟨ς,κ, ϑ⟩ if U (G) = intσ(cl

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

(3) τ
(
Fl(intσ(cl

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))

)
≥ ⟨ς,κ, ϑ⟩ if σ(U (G)) ≥ ⟨ς,κ, ϑ⟩.

Proof. (1) =⇒ (2) If U (G) = intσ(cl
∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), then σ(U (G)) ≥

⟨ς,κ, ϑ⟩. By Theorem 4.1(2), Fl(U (G)) ⊆ intτ
(
Fl(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)

)
, ⟨ς,κ, ϑ⟩) =

intτ
(
Fl(U (G)), ⟨ς,κ, ϑ⟩

)
. Thus, τ

(
Fl (U (G))

)
≥ ⟨ς,κ, ϑ⟩.

(2) ⇔ (3) Obvious.
(3) =⇒ (1) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then by (3) and U (G) ⊆ intσ(cl
∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩),

τ
(
Fl(intσ(cl

∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))

)
≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl (U (G)) ⊆ Fl(intσ(cl
∗
σ (U (G) , ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). Thus, F is TPF lA

LP -continuous.

The following theorems are similarly proved as the proof of Theorem 4.3.



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 15 of 25

Theorem 4.4. For a TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF lA LP -continuous.
(2) τ (Ⅎ Fu (U (G))) ≥ ⟨ς,κ, ϑ⟩, if U (G) = clσ(int

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

(3) τ (Ⅎ Fu(clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩ if σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩.

Theorem 4.5. For a NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF uA LP -continuous.
(2) τ (Fu (U (G))) ≥ ⟨ς,κ, ϑ⟩, if U (G) = intσ(cl

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

(3) τ (Fu(intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩ if σ(U (G)) ≥ ⟨ς,κ, ϑ⟩.

Theorem 4.6. For a NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:
(1) F is TPF uA LP -continuous.
(2) τ

(
Ⅎ Fl (U (G))

)
≥ ⟨ς,κ, ϑ⟩, if U (G) = clσ(int

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

(3) τ
(
Ⅎ Fl(clσ(int

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))

)
≥ ⟨ς,κ, ϑ⟩ if σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩.

Theorem 4.7. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a TPFM . Then, F is TPF lA LP -continuous
iff
clτ (F

u (U (G)) , ⟨ς,κ, ϑ⟩) ⊆ Fu(clσ(U (G) , ⟨ς,κ, ϑ⟩)) for any U (G) ∈
(
I3
)Υ×G with

U (G) ⊆ clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

Proof. (⇒) Let F be a TPF lA LP -continuous. Then for any U (G) ∈
(
I3
)Υ×G with

U (G) ⊆ clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) = K (G) (say) where K (G) = clσ(int

∗
σ (K (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

By Theorem 3.6, τ (Ⅎ Fu (K (G))) ≥ ⟨ς,κ, ϑ⟩, and thus
clτ (F

u (U (G)) , ⟨ς,κ, ϑ⟩) ⊆ clτ (F
u (K (G)) , ⟨ς,κ, ϑ⟩) = Fu(clσ(int

∗
σ (K (G), ⟨ς,κ, ϑ⟩) ,

⟨ς,κ, ϑ⟩))
⊆ Fu(clσ(U (G) , ⟨ς,κ, ϑ⟩)).

(⇐) Let U (G) ∈
(
I3
)Υ×G with U (G) = clσ(int

∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). Then,

U (G) ⊆ clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) and

clτ (F
u (U (G)) , ⟨ς,κ, ϑ⟩) ⊆ Fu(clσ(U (G) , ⟨ς,κ, ϑ⟩)) = Fu (U (G)) . Therefore, we obtain

τ (Ⅎ Fu (U (G))) ≥ ⟨ς,κ, ϑ⟩. Thus by Theorem 4.1, F is TPF lA LP -continuous.

The following theorem is similarly proved as the proof of Theorem 4.7.

Theorem 4.8. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a NTPFM . Then F is TPF uA LP -
continuous iff
clτ

(
Fl (U (G)) , ⟨ς,κ, ϑ⟩

)
⊆ Fl(clσ(U (G) , ⟨ς,κ, ϑ⟩)) for any U (G) ∈

(
I3
)Υ×G with

U (G) ⊆ clσ(int
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

5. Temporal picture fuzzy weakly continuous multifunctions

Definition 5.1. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a TPFM , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.
Then, F is called:



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 16 of 25

(1) TPF uW LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈
Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G,

τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G)∩D (F) ⊆ Fu(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)).
(2) TPF lW LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈

Fl(U (G)) for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G,

τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ⊆ Fl(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)).
(3) TPF uW LP -continuous(resp. TPF lW LP -continuous) iff it is TPF uW LP -continuous(resp.

TPF lW LP -continuous) at every fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F).

Remark 5.1. (1) If F is NTPFM , then F is TPF uW LP -continuous at a fuzzy point
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G)) for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists

G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G such that G (G) ⊆ Fu(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)).
(2) TPF uA (resp. TPF lA) LP -continuity ⇒ TPF uW (resp. TPF lW ) LP -continuity

⇒ TPF uW (resp. TPF lW )-continuity.
(3) TPF uW (resp. TPF lW ) LP0-continuity ⇔ TPF uW (resp. TPF lW )-continuity.

Theorem 5.1. A TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF lW LP -continuous iff
Fl(U (G)) ⊆ intτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for each U (G) ∈
(
I3
)Υ×Gwith σ(U (G)) ≥

⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

Proof. (⇒) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×Gwith σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then, there exists G (G) ∈
(
I3
)ℵ×G, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that
G (G) ⊆ Fl(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)). Thus, ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) ⊆ intτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)
and hence Fl (U (G)) ⊆ intτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).
(⇐) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F),U (G) ∈

(
I3
)Υ×Gwith σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈

Fl(U (G)). Then, ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)) ⊆ intτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus,

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) ⊆ intτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fl(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)).

Hence, F is TPF uW LP -continuous.

The following theorem is similarly proved as the proof of Theorem 5.1.

Theorem 5.2. A NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uW LP -continuous iff
Fu((U (G)) ⊆ intτ (F

u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for each U (G) ∈
(
I3
)Υ×G with

σ(U (G)) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

The following examples shows that generally a TPF uW LP -continuous and TPF lW
LP -continuous (resp. a TPF uW continuous and TPF lW continuous) multifunction need
not be either a TPF uA LP -continuous (resp. TPF uW LP -continuous) multifunction or
TPF lA LP -continuous (resp. TPF lW LP -continuous) multifunction.



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 17 of 25

Example 5.1. From the Example 3.2,
F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uW (resp. TPF lW )-continuous but is not TPF uW

(resp. TPF lW ) LP -continuous because

Fu(U1 (G)) = G2 (G) ⊆ intτ (F
u(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) =

♯ (G) ,
Fl(U1 (G)) = G2 (G)⊆ intτ

(
Fl(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) = ♯ (G) ,

but
Fu(U1 (G)) = G2 (G) ⊈ intτ (F

u(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) =
♭ (G) ,

Fl(U1 (G)) = G2 (G)⊈ intτ
(
Fl(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) = ♭ (G) ,

Example 5.2. Let ℵ = {ϱ1, ϱ2}, Υ = {ζ1, ζ2, }, G = {g1, g2} and F : ℵ ↬ Υ be a TPFM
defined by
ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) as:

ΨF(⟨ϱ, g⟩ , ⟨ζ, g⟩) ⟨ζ1, g1⟩ ⟨ζ1, g2⟩ ⟨ζ2, g1⟩ ⟨ζ2, g2⟩
⟨ϱ1, g1⟩ ⟨0.2, 0.3, 0.4⟩ ⟨1, 0, 0⟩ ⟨0.2, 0.5, 0.1⟩ ⟨0.6, 0.2, 0.1⟩
⟨ϱ1, g2⟩ ⟨0.4, 0.2, 0.4⟩ ⟨1, 0, 0⟩ ⟨0.3, 0.6, 0.1⟩ ⟨0.3, 0.3, 0.3⟩
⟨ϱ2, g1⟩ ⟨0.2, 0.2, 0.2⟩ ⟨0.1, 0.1, 0.1⟩ ⟨1, 0, 0⟩ ⟨0.44, 0.2, 0.1⟩
⟨ϱ2, g2⟩ ⟨0.5, 0.1, 0.2⟩ ⟨0.1, 0.1, 0.1⟩ ⟨0.2, 0.3, 0.5⟩ ⟨1, 0, 0⟩ .

Define temporal picture fuzzy topologies τ :
(
I3
)ℵ×G → I3, σ:

(
I3
)Υ×G → I3 , and

temporal picture fuzzy ideal LP :
(
I3
)Υ×G → I3 as:

τ(G (G)) =


⟨1, 0, 0⟩ , G (G) ∈ {♭ (G) , ♯ (G)}

⟨0.6, 0.2, 0.2⟩ , G (G) = G1 (G)

⟨0, 1, 0⟩ , o.w,

σ(U(G)) =


⟨1, 0, 0⟩ , U (G) ∈ {♭ (G) , ♯ (G)}

⟨0.4, 0.45, 0.15⟩ , U(G) = U1(G)

⟨0, 1, 0⟩ , o.w.

LP (U(G)) =



⟨1, 0, 0⟩ , U(G) = ♭ (G)

⟨0.3, 0.1, 0.6⟩ ,
{

⟨⟨ζ, g1⟩ , 0.44, 0.4, 0.16⟩ ,
⟨⟨ζ, g2⟩ , 0.44, 0.41, 0.15⟩ ,

, ζ ∈ Υ

}
⊆ U(G) ⊂ ♯ (G)

⟨0, 1, 0⟩ , o.w,

where G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.44, 0.41, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.44, 0.41, 0⟩ ,

, ϱ ∈ ℵ
}

and U1 (G) =

{
⟨⟨ζ, g1⟩ , 0.4, 0.44, 0.2⟩ ,
⟨⟨ζ, g2⟩ , 0.41, 0.44, 0.15⟩ ,

, ζ ∈ Υ

}
.

Then, F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uW (resp. TPF lW ) LP -continuous but is not
TPF uA (resp. TPF lA)-continuous because



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 18 of 25

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.4, 0.44, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.4, 0.44, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ (F
u(cl∗σ (U1 (G) , ⟨0.4, 0.45, 0.15⟩)) , ⟨0.4, 0.45, 0.15⟩) = G1 (G) ,

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.41, 0.44, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.41, 0.44, 0⟩ ,

, ϱ ∈ ℵ
}

⊆ intτ
(
Fl(cl∗σ (U1 (G) , ⟨0.4, 0.45, 0.15⟩)

)
, ⟨0.4, 0.45, 0.15⟩) = G1 (G) ,

but

Fu(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.4, 0.44, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.4, 0.44, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ (F
u(intσ(cl

∗
σ(U1 (G), ⟨0.4, 0.45, 0.15⟩), ⟨0.4, 0.45, 0.15⟩)) , ⟨0.4, 0.45, 0.15⟩) = ♭ (G) ,

Fl(U1 (G)) =

{
⟨⟨ϱ, g1⟩ , 0.41, 0.44, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.41, 0.44, 0⟩ ,

, ϱ ∈ ℵ
}

⊈ intτ
(
Fl(intσ(cl

∗
σ(U1 (G), ⟨0.4, 0.45, 0.15⟩), ⟨0.4, 0.45, 0.15⟩)

)
, ⟨0.4, 0.45, 0.15⟩) = ♭ (G) .

Theorem 5.3. A TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF lW LP -continuous iff
clτ (F

u(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu(U (G)) for each U (G) ∈
(
I3
)Υ×G with

σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

Proof. (⇒) Let U (G) ∈
(
I3
)Υ×Gwith σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩ . Then by Theorem 4.7,

Ⅎ Fu(U (G)) = Fl(Ⅎ U (G)) ⊆ intτ (F
l(cl∗σ (Ⅎ U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)

= Ⅎ clτ (F
u(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).

Thus, clτ (Fu(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu(U (G)).
(⇐) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈

(
I3
)Υ×Gwith σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and ξt ∈

Fl(U (G)). Then,

Ⅎintτ (Fl(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) = clτ (F
u(int∗σ (Ⅎ U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)

⊆ Fu(Ⅎ U (G)) = Ⅎ Fl(U (G)),

and hence Fl(U (G)) ⊆ intτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, it is TPF lW LP -

continuous.

The following theorem is similarly proved as the proof of Theorem 5.3.

Theorem 5.4. A NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uW LP -continuous iff
clτ (F

l(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fl(U (G)) for each U (G) ∈
(
I3
)Υ×G with σ(Ⅎ

U (G)) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

Theorem 5.5. If F : (ℵ, τ) ↬ (Υ, σ,LP ) is NTPF uW LP -continuous and
F (G (G)) ⊆ intσ(cl

∗
σ(F (G (G)) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) for each G (G) ∈

(
I3
)ℵ×G then F is

TPF uA LP -continuous.

Proof. Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G,σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈

Fu(U (G)). Then, there exists G (G) ∈
(
I3
)ℵ×G with τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 19 of 25

G (G) such that
G (G) ⊆ Fu(cl∗σ(U (G) , ⟨ς,κ, ϑ⟩)), then F (G (G)) ⊆ F (Fu(cl∗σ(U (G), ⟨ς,κ, ϑ⟩))) ⊆ cl∗σ(U (G) , ⟨ς,κ, ϑ⟩).
Since F (G (G)) ⊆ intσ(cl

∗
σ(F (G (G)) , ⟨ς,κ, ϑ⟩),

⟨ς,κ, ϑ⟩) ⊆ intσ(cl
∗
σ(U (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), hence

G (G) ⊆ Fu (F (G (G))) ⊆ Fu (intσ(cl
∗
σ(U (G), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)). Then, F is TPF uA

LP -continuous.

Theorem 5.6. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a TPF lW LP -continuous. Then,
Fl(U (G)) ⊆ intτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for any U (G) ∈
(
I3
)Υ×G with

U (G) ⊆ intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

Proof. Let F be a TPF lW LP -continuous and U (G) ∈
(
I3
)Υ×G with

U (G) ⊆ intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩).

Then, if ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)) ⊆ Fl(intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), there exists

G (G) ∈
(
I3
)ℵ×G,

τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that

G (G) ⊆ (Fl(cl∗σ(intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)) ⊆ Fl(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)).

Thus, G (G) ⊆ intτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) and

Fl(U (G)) ⊆ intτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).

The following theorem is similarly proved as the proof of Theorem 5.6.

Theorem 5.7. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a NTPF uW LP -continuous . Then,
Fu((U (G)) ⊆ intτ (F

u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for any U (G) ∈
(
I3
)Υ×G with

U (G) ⊆ intσ(cl
∗
σ (U (G), ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.

6. Temporal picture fuzzy almost weakly continuous multifunctions

Definition 6.1. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a TPFM , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1.
Then, F is called:

(1) TPF uAW LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈
Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G)∩D (F) ⊆ clτ (F

u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).
(2) TPF lAW LP -continuous at a fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈

Fl(U (G)) for each U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ there exists G (G) ∈

(
I3
)ℵ×G

, τ(G (G)) ≥
⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ⊆ clτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).
(3) TPF uAW LP -continuous(resp. TPF lAW LP -continuous) iff it is TPF uAW LP -

continuous
(resp. TPF lAW LP -continuous) at every fuzzy point ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 20 of 25

Remark 6.1. (1) If F is NTPFM , then F is TPF uAW LP -continuous at a fuzzy point
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F) iff ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fu(U (G)) for each U (G) ∈

(
I3
)Υ×G, σ(U (G)) ≥

⟨ς,κ, ϑ⟩ there exists G (G) ∈
(
I3
)ℵ×G,

τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that G (G) ⊆ clτ (F
u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩).

(2) TPF uW (resp. TPF lW ) LP -continuity ⇒ TPF uAW (resp. TPF lAW ) LP -
continuity ⇒ TPF uAW (resp. TPF lAW )-continuity.

(3) TPF uAW (resp. TPF lAW ) LP0-continuity ⇔ TPF uAW (resp. TPF lAW )-
continuity.

Theorem 6.1. For a TPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0,κ ∈ I1

and ϑ ∈ I1,the following statements are equivalent:
(1) F is TPF lAW LP -continuous.
(2) Fl((U (G)) ⊆ intτ (clτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩),if σ(U (G)) ≥
⟨ς,κ, ϑ⟩ .

(3) clτ (intτ (F
u(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Fu (U (G)), if σ(Ⅎ U (G)) ≥

⟨ς,κ, ϑ⟩ .

Proof. (1) =⇒ (2) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G,σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then, there exists G (G) ∈
(
I3
)ℵ×G

, τ(G (G)) ≥ ⟨ς,κ, ϑ⟩ and
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) such that
G (G) ⊆ clτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus,
⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ G (G) ⊆ intτ (clτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), and hence
Fl (U (G)) ⊆ intτ (clτ (F

l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩).
(2) =⇒ (3) Let U (G) ∈

(
I3
)Υ×G with σ(Ⅎ U (G)) ≥ ⟨ς,κ, ϑ⟩. Then by (2) ,

Ⅎ Fu (U (G)) = Fl(Ⅎ U (G)) ⊆ intτ (clτ (F
l(cl∗σ (Ⅎ U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)

= Ⅎ clτ (intτ (F
u(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ,

thus clτ (intτ (F
u(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ⊆ Fu (U (G)) .

(3) =⇒ (1) Let ⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ D (F), U (G) ∈
(
I3
)Υ×G, σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ Fl(U (G)). Then by (3), we have

Ⅎ intτ (clτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)

= clτ (intτ (F
u(int∗σ (Ⅎ U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ⊆ Fu (Ⅎ U (G)) = Ⅎ Fl(U (G)),

and hence Fl((U (G)) ⊆ intτ (clτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). Therefore,

⟨ϱ, g⟩⟨ς,κ,ϑ⟩ ∈ intτ (clτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)

⊆ clτ (F
l(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, F is TPF lAWLP -continuous.

The following theorem is similar to Theorem 4.5.

Theorem 6.2. For a NTPFM F : (ℵ, τ) ↬ (Υ, σ,LP ), U (G) ∈
(
I3
)Υ×G, ς ∈ I0, κ ∈ I1

and ϑ ∈ I1, the following statements are equivalent:



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 21 of 25

(1) F is TPF uAW LP -continuous.
(2) Fu((U (G)) ⊆ intτ (clτ (F

u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩),if σ(U (G)) ≥
⟨ς,κ, ϑ⟩ .

(3) clτ
(
intτ (F

l(int∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩
)
⊆ Fl (U (G)), if σ(Ⅎ U (G)) ≥

⟨ς,κ, ϑ⟩ .

The following example shows that generally a TPF uAW continuous and TPF lAW con-
tinuous (resp. TPF uAW LP -continuous and TPF lAW LP -continuous) need not be either
a TPF uAW LP -continuous (resp. TPF uW LP -continuous) or TPF lAW LP -continuous
(resp. TPF lW LP -continuous).

Example 6.1. From the Example 4.1, define temporal picture fuzzy topologies τ :
(
I3
)ℵ×G →

I3 and temporal picture fuzzy ideal LP :
(
I3
)Υ×G → I3 as follows:

τ(G (G)) =


⟨1, 0, 0⟩ , G (G) ∈ {♭ (G) , ♯ (G)}

⟨0.6, 0.2, 0.2⟩ , G (G) = G1 (G)

⟨0, 1, 0⟩ , o.w,

LP (U(G)) =



⟨1, 0, 0⟩ , U(G) = ♭ (G)

⟨0.55, 0.25, 0.2⟩ , ♭ (G) ⊂ U(G) ⊆
{

⟨⟨ζ, g1⟩ , 0.4, 0.4, 0.1⟩ ,
⟨⟨ζ, g2⟩ , 0.44, 0.41, 0.1⟩ ,

, ζ ∈ Υ

}
⟨0, 1, 0⟩ , o.w,

where G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.3, 0.5, 0.2⟩ ,
⟨⟨ϱ, g2⟩ , 0.2, 0.6, 0.1⟩ ,

, ϱ ∈ ℵ
}

Then, (1) F : (ℵ, τ) ↬ (Υ, σ,LP ) is TPF uAW (resp. TPF lAW )-continuous but is not
TPF uAW (resp. TPF lAW ) LP -continuous because

Fu(U1 (G)) = G2 (G)
⊆ intτ (clτ (F

u(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩) = ♯ (G) ,
Fl(U1 (G)) = G2 (G)
⊆ intτ (clτ

(
Fl(clσ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩) = ♯ (G) ,

but
Fu(U1 (G)) = G2 (G)
⊈ intτ (clτ (F

u(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)

=

{
⟨⟨ϱ, g1⟩ , 0.3, 0.5, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.2, 0.6, 0⟩ ,

, ϱ ∈ ℵ
}
,

Fl(U1 (G)) = G2 (G)
⊈ intτ (clτ

(
Fl(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)

=

{
⟨⟨ϱ, g1⟩ , 0.3, 0.5, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.2, 0.6, 0⟩ ,

, ϱ ∈ ℵ
}
,



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 22 of 25

(2) For G1 (G) =

{
⟨⟨ϱ, g1⟩ , 0.5, 0.4, 0⟩ ,
⟨⟨ϱ, g2⟩ , 0.5, 0.4, 0⟩ ,

, ϱ ∈ ℵ
}

then F : (ℵ, τ) ↬ (Υ, σ,LP ) is

TPF uAW (resp.
TPF lAW ) LP -continuous but is not TPF uW (resp. TPF lW ) LP -continuous because
Fu(U1 (G)) = G2 (G)⊆ intτ (clτ (F

u(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)
= ♯ (G),
Fl(U1 (G)) = G2 (G)⊆ intτ (clτ

(
Fl(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩), ⟨0.31, 0.31, 0.38⟩)

= ♯ (G) , but
Fu(U1 (G)) = G2 (G) ̸⊆ intτ (F

u(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)) , ⟨0.31, 0.31, 0.38⟩) =
♭ (G),
Fl(U1 (G)) = G2 (G) ̸⊆ intτ

(
Fl(cl∗σ(U1 (G), ⟨0.31, 0.31, 0.38⟩)

)
, ⟨0.31, 0.31, 0.38⟩) = ♭ (G) .

Theorem 6.3. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a NTPFM , F be TPF uAW LP -continuous
and TPF lA LP -continuous. Then, F is TPF uW LP -continuous.

Proof. Let U (G) ∈
(
I3
)Υ×Gwith σ(U (G)) ≥ ⟨ς,κ, ϑ⟩ and F be TPF uAW LP -

continuous. Then by Theorem 6.1(1),

Fu(U (G)) ⊆ intτ (clτ (F
u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩).

Since clσ(U (G) , ⟨ς,κ, ϑ⟩) = clσ(int
∗
σ(clσ(U (G) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), it follows

from Theorem 4.3(2) that τ (Ⅎ Fu (clσ(U (G), ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩, then τ (Ⅎ Fu (cl∗σ(U (G), ⟨ς,κ, ϑ⟩))) ≥
⟨ς,κ, ϑ⟩, and Fu(U (G)) ⊆ intτ (F

u(cl∗σ (U (G), ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus by Theorem 5.4,
F is TPF uW LP -continuous.

The following theorem is similarly proved as the proof of Theorem 6.3.

Theorem 6.4. Let F : (ℵ, τ) ↬ (Υ, σ,LP ) be a NTPFM , F be TPF lAW LP -continuous
and TPF uA LP -continuous. Then F is TPF lW LP -continuous.

An applications M, N, idℵ×G :
(
I3
)ℵ×G × I3 →

(
I3
)ℵ×G are operators on ℵ and W,

V, idΥ×G :
(
I3
)Υ×G × I3 →

(
I3
)Υ×G are operators on Υ.

Definition 6.2. (1) Let F : (ℵ, τ,LP ) ↬ (Υ, σ) be a TPFM . Then, F is TPF l (M,N,W,V,LP )-
continuous iff
for every U (G) ∈

(
I3
)Υ×G, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1,

LP [M(Fl (V(U (G) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)⊼N(Fl (W(U (G) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)] ≥ σ(U (G)).

(2) Let F : (Ξ, τ,LP
1 ) ↬ (Υ, σ,LP

2 ) be a NTPFM . Then F is TPF u (M,N,W,V,LP )-
continuous iff
for every U (G) ∈

(
I3
)Υ×G, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1,

LP [M(Fu (V(U (G) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)⊼N(Fu (W(U (G) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩)] ≥ σ(U (G)).



D. Shi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6155 23 of 25

Remark 6.2. 1. TPF l (resp. NTPF u) LP -continuous multifunction ⇔ TPF l (resp.
NTPF u) (idℵ×G, intτ (Φτ ), intσ, idΥ×G, LP0)-continuous multifunction.

2. TPF lA (resp. NTPF uA) LP -continuous multifunction⇔ TPF l (resp. NTPF u)
(idℵ×G, intτ , intσ (cl∗σ), idΥ×G, LP0)-continuous multifunction.

3. TPF lW (resp. NTPF uW ) LP -continuous multifunction⇔ TPF l (resp. NTPF u)
(idℵ×G, intτ , cl∗σ, idΥ×G, LP0)-continuous multifunction.

4. TPF lAW (resp. NTPF uAW ) LP -continuous multifunction⇔ TPF l (resp. NTPF u)
(idℵ×G, intτ (clτ ), cl∗σ, idΥ×G, LP0)-continuous multifunction.

7. Conclusion

This paper submitted the notions of TPF u or TPF l-continuous, TPF u or TPF l al-
most continuous, TPF u or TPF l weakly continuous and TPF u or TPF l almost weakly
continuous multifunctions depending on a TPF -ideal. Some characterizations of these
types of TPF -continuous multifunctions are proved, and many examples are submitted
to explain the allowed implications between these types of TPF continuity. That is, the
variety of continuity of TPF -multifunctions based on TPF -ideals and the implications in
between are meaningful and have been discussed in detail. In future work, we will gener-
alize these notions to wider forms of TPF -semi continuity. Also, we will try to study the
variety of TPF -continuity in the fuzzy soft set theory using special operators.

Authors Contributions: Resources, Methodology and Funding, Shi; Validation and
Formal analysis, Abbas and Shi; Reviewing and Investigation the final version, Abbas
and Ibedou; Writing-original draft, Abu_Shugair and Ibedou; Visualization, Ibedou and
Abbas. The authors all confirmed this published version of the manuscript.

Conflicts of interest: The authors declare that they have no conflict of interest.
Data Availability Statement: The data sets used and/or analyzed during the current
study are available from the corresponding author upon reasonable request.

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