θ(⋆)-quasi continuity for multifunctions
CHAWALIT BOONPOK
Mathematics and Applied Mathematics Research Unit
Department of Mathematics, Faculty of Science, Mahasarakham University
Maha Sarakham, 44150
THAILAND
Abstract: Our purpose is to introduce the concepts of upper and lower θ(⋆)-quasi continuous multifunctions.
Several characterizations of upper and lower θ(⋆)-quasi continuous multifunctions are investigated.
Key–Words: upper θ(⋆)-quasi continuous multifunction, lower θ(⋆)-quasi continuous multifunction
Received: June 14, 2021. Revised: March 17, 2022. Accepted: April 15, 2022. Published: May 20, 2022.
1 Introduction
The notion of continuity is an important concept in
general topology as well as all branches of mathemat-
ics. This concept has been extended to the setting
multifunctions and has been generalized by weaker
forms of open sets such as α-open sets [15], semi-
open sets [13], preopen sets [14], β-open sets [2]
or semi-preopen sets [4]. Levine [13] introduced
the concept of semi-continuity in topological spaces.
Arya and Bhamini [5] introduced and studied the
concept of θ-semi-continuity as a generalization of
semi-continuity. Noiri [18] and Jafari and Noiri [10]
have further investigated several characterizations of
θ-semi-continuity. Popa and Noiri [19] introduced
and studied the notions of upper and lower θ-quasi
continuous multifunctions. The present authors [16]
obtained new characterizations of upper and lower θ-
quasi continuous multifunctions. Topological ideals
have played an important role in topology. The con-
cept of ideal topological spaces was introduced and
studied by Kuratowski [12] and Vaidyanathaswamy
[20]. Jankovi´
c and Hamlett [11] introduced the notion
of I-open sets in ideal topologial spaces. Abd El-
Monsef et al. [1] further investigated I-open sets and
I-continuous functions. Later, several authors stud-
ied ideal topological spaces giving several convenient
definitions. Some authors obtained decompositions of
continuity. For instance, Ac¸ikg¨
oz et al. [3] introduced
and investigated the notions of weakly-I-continuous
and weak⋆-I-continuous functions in ideal topolog-
ical spaces. Hatir and Noiri [9] introduced the no-
tions of semi-I-open sets, α-I-open sets and β-I-
open sets via idealization and using these sets ob-
tained new decompositions of continuity. In [7], the
author introduced and investigated the concepts of al-
most quasi ⋆-continuous multifunctions and weakly
quasi ⋆-continuous multifunctions. The purpose of the
present paper is to introduce the notions of upper and
lower θ(⋆)-quasi continuous multifunctions. More-
over, several characterizations of θ(⋆)-quasi continu-
ous multifunctions are discussed.
2 Preliminaries
Throughout the present paper, spaces (X, τ )and
(Y, σ)(or simply Xand Y) always mean topological
spaces on which no separation axioms are assumed
unless explicitly stated. Let Abe a subset of a topo-
logical space (X, τ). The closure of Aand the inte-
rior of Aare denoted by Cl(A)and Int(A), respec-
tively. An ideal Ion a topological space (X, τ )is
a nonempty collection of subsets of Xsatisfying the
following properties: (1) A∈Iand B⊆Aimply
B∈I; (2) A∈Iand B∈Iimply A∪B∈I.
A topological space (X, τ )with an ideal Ion Xis
called an ideal topological space and is denoted by
(X, τ, I). For an ideal topological space (X, τ, I)
and a subset Aof X,A⋆(I)is defined as follows:
A⋆(I) ={x∈X:U∩A∈ I
for every open neighbourhood Uof x}.
In case there is no chance for confusion, A⋆(I)is
simply written as A⋆. In [12], A⋆is called the local
function of Awith respect to Iand τ. Observe addi-
tionally that Cl⋆(A) = A⋆∪Adefines a Kuratowski
closure operator for a topology τ⋆(I)finer than τ,
generated by the base
B(I, τ) = {U−I|U∈τand I∈I}.
However, B(I, τ )is not always a topology [20]. A
subset Ais said to be ⋆-closed [11] if A⋆⊆A. The
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interior of a subset Ain (X, τ⋆(I)) is denoted by
Int⋆(A).
A subset Aof an ideal topological space
(X, τ, I)called R-I⋆-open (resp. I⋆-preopen) [8]
if A=Int⋆(Cl⋆(A)) (resp. A⊆Int⋆(Cl⋆(A))).
The complement of a R-I⋆-open (resp. I⋆-preopen)
set is called R-I⋆-closed (resp. I⋆-preclosed). A
subset Aof an ideal topological space (X, τ, I)is
called semi-I⋆-open (resp. semi-I⋆-preopen) [7] if
A⊆Cl⋆(Int⋆(A)) (resp. A⊆Cl⋆(Int⋆(Cl⋆(A)))).
The complement of a semi-I⋆-open (resp. semi-I⋆-
preopen) set is called semi-I⋆-closed (resp. semi-
I⋆-preclosed). For a subset Aof an ideal topo-
logical space (X, τ, I), the intersection of all semi-
I⋆-closed sets containing Ais called the semi-I⋆-
closure [7] of Aand is denoted by sClI⋆(A). The
union of all semi-I⋆-open sets contained in Ais
called the semi-I⋆-interior [7] of Aand is denoted
by sIntI⋆(A).
Lemma 1. [7] For a subset Aof an ideal topological
space (X, τ, I),x∈sClI⋆(A)if and only if
U∩A=∅
for every semi-I⋆-open set Ucontaining x.
Let Abe a subset of an ideal topological space
(X, τ, I). A point xin an ideal topological space
(X, τ, I)is called a ⋆θ-cluster point [7] of Aif
Cl⋆(U)∩A=∅for every ⋆-open set Uof Xcontain-
ing x. The set of all ⋆θ-cluster points of Ais called the
⋆θ-closure [7] of Aand is denoted by ⋆θCl(A). A sub-
set Bof an ideal topological space (X, τ, I)is said
to be ⋆θ-closed if ⋆θCl(B) = B. The complement of
a⋆θ-closed set is said to be ⋆θ-open.
Lemma 2. [7] For a subset Aof an ideal topological
space (X, τ, I), the following properties hold:
(1) If Ais ⋆-open in X, then Cl⋆(A) = ⋆θCl(A).
(2) ⋆θCl(A)is ⋆-closed in X.
The family of all semi-I⋆-open sets of an ideal
topological space (X, τ, I)is denoted by SI⋆O(X).
Let Abe a subset of an ideal topological space
(X, τ, I). The semi-θ(⋆)-closure of A,⋆θsCl(A)and
the semi-θ(⋆)-interior of A,⋆θsInt(A)are defined as
follows:
⋆θsCl(A) ={x∈X|A∩sClI⋆(U)=∅
for every U∈SI⋆O(X, x)},
⋆θsInt(A) ={x∈X|sClI⋆(U)⊆Afor some
U∈SI⋆O(X, x)}
where
SI⋆O(X, x) = {U|x∈Uand U∈SI⋆O(X)}.
A subset Aof an ideal topological space
(X, τ, I)is called semi-θ(⋆)-closed if A=
⋆θsCl(A).
By a multifunction F:X→Y, we mean a point-
to-set correspondence from Xinto Y, and we always
assume that F(x)=∅for all x∈X. For a multifunc-
tion F:X→Y, following [6] we shall denote the
upper and lower inverse of a set Bof Yby F+(B)
and F−(B), respectively, that is,
F+(B) = {x∈X|F(x)⊆B}
and F−(B) = {x∈X|F(x)∩B=∅}. In partic-
ular, F−(y) = {x∈X|y∈F(x)}for each point
y∈Y. For each A⊆X,F(A) = ∪x∈AF(x). Then
Fis said to be surjection if F(X) = Y, or equiva-
lent, if for each y∈Ythere exists x∈Xsuch that
y∈F(x)and Fis called injection if x=yimplies
F(x)∩F(y) = ∅.
3 Characterizations of upper and
lower θ(⋆)-quasi continuous multi-
functions
In this section, we introduce the notions of upper and
lower θ(⋆)-quasi continuous multifunctions. More-
over, some characterizations of upper and lower θ(⋆)-
quasi continuous multifunctions are discussed.
Definition 3. A multifunction F: (X, τ, I)→
(Y, σ, J)is said to be:
(1) upper θ(⋆)-quasi continuous if, for each x∈X
and each ⋆-open set Vof Ycontaining F(x),
there exists a semi-I⋆-open set Uof Xcontain-
ing xsuch that F(sClI⋆(U)) ⊆Cl⋆(V);
(2) lower θ(⋆)-quasi continuous if, for x∈Xand
each ⋆- open set Vof Ysuch that F(x)∩V=∅,
there exists a semi-I⋆-open set Uof Xcontain-
ing xsuch that F(z)∩Cl⋆(V)=∅for every
z∈sClI⋆(U).
Example 4. Let X={1,2,3}with a topology
τ={∅,{1},{2},{1,2}, X}and an ideal I=
{∅,{3}}. Let Y={a, b, c}with a topology σ=
{∅,{a},{a, b}, Y }and an ideal J={∅,{c}}. A
multifunction F: (X, τ, I)→(Y, σ, J)is de-
fined as follows: F(1) = {a},F(2) = {c}and
F(3) = {a, b}. Then, Fis upper θ(⋆)-quasi continu-
ous.
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Example 5. Let X={a, b, c}with a topology
τ={∅,{c}, X}and an ideal I={∅,{c}}.
Let Y={−1,0,1,2}with a topology σ=
{∅,{−1,0},{1,2}, Y }and an ideal J={∅}. De-
fine a multifunction F: (X, τ, I)→(Y, σ, J)as
follows: F(a) = {−1},F(b) = {0}and F(c) =
{1,2}. Then, Fis lower θ(⋆)-quasi continuous.
The following theorem gives some characteriza-
tions of upper θ(⋆)-quasi continuous multifunctions.
Theorem 6. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis upper θ(⋆)-quasi continuous;
(2) ⋆θsCl(F−(Int⋆(⋆θCl(B)))) ⊆F−(⋆θCl(B))
for every subset Bof Y;
(3) ⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)) for
every ⋆-open set Vof Y;
(4) ⋆θsCl(F−(Int⋆(K))) ⊆F−(K)for every R-
J⋆-closed set Kof Y;
(5) F+(V)⊆⋆θsInt(F+(Cl⋆(V))) for every ⋆-
open set Vof Y;
(6) ⋆θsCl(F−(Int⋆(K))) ⊆F−(K)for every ⋆-
closed set Kof Y;
(7) ⋆θsCl(F−(V)) ⊆F−(Cl⋆(V)) for every ⋆-open
set Vof Y.
Proof. (1) ⇒(2): Let Bbe any subset of Y. Suppose
that x∈ F−(⋆θCl(B)). Then, x∈X−F−(⋆θCl(B))
and F(x)⊆Y−⋆θCl(B). Since ⋆θCl(B)is ⋆-closed
in Y, there exists a semi-I⋆-open set Uof Xcon-
taining xsuch that
F(sClI⋆(U)) ⊆Cl⋆(Y−⋆θCl(B))
=Y−Int⋆(⋆θCl(B)).
Thus, F(sClI⋆(U)) ∩Int⋆(⋆θCl(B)) = ∅and hence
sClI⋆(U)∩F−(Int⋆(⋆θCl(B))) = ∅. This shows that
x∈ ⋆θsCl(F−(Int⋆(⋆θCl(B)))). Consequently, we
obtain ⋆θsCl(F−(Int⋆(⋆θCl(B)))) ⊆F−(Clθ⋆(B)).
(2) ⇒(3): This is obvious since Cl⋆(V) =
⋆θCl(V)for every ⋆-open set Vof Y.
(3) ⇒(4): Let Kbe any R-J⋆-closed set of Y.
By (3), we have
⋆θsCl(F−(Int⋆(K)))
=⋆θsCl(F−(Int⋆(Cl⋆(Int⋆(K)))))
⊆F−(Cl⋆(Int⋆(K)))
=F−(K).
(4) ⇒(5): Let Vbe any ⋆-open set of Y. Then,
we have
X−⋆θsInt(F+(Cl⋆(V)))
=⋆θsCl(X−F+(Cl⋆(V)))
=⋆θsCl(F−(Y−Cl⋆(V))),
Y−Cl⋆(V) = Int⋆(Y−Cl⋆(V))
⊆Int⋆(Y−Int⋆(Cl⋆(V)))
and Y−Int⋆(Cl⋆(V)) is R-J⋆-closed in Y. By (4),
⋆θsCl(F−(Int⋆(Y−Int⋆(Cl⋆(V)))))
⊆F−(Y−Int⋆(Cl⋆(V)))
=X−F+(Int⋆(Cl⋆(V)))
⊆X−F+(V).
Therefore, F+(V)⊆⋆θsInt(F+(Cl⋆(V))).
(5) ⇒(6): Let Kbe any ⋆-closed set of Y. By
(5), we have
X−F−(K) = F+(Y−K)
⊆⋆θsInt(F+(Cl⋆(Y−K)))
=⋆θsInt(F+(Y−Int⋆(K)))
=⋆θsInt(X−F−(Int⋆(K)))
=X−⋆θsCl(F−(Int⋆(K)))
and hence ⋆θsCl(F−(Int⋆(K))) ⊆F−(K).
(6) ⇒(7): Let Vbe any ⋆-open set of Y. Then,
we have Cl⋆(V)is ⋆-closed and by (6),
⋆θsCl(F−(V)) ⊆⋆θsCl(F−(Int⋆(Cl⋆(V))))
⊆F−(Cl⋆(V)).
(7) ⇒(1): Let x∈Xand Vbe any ⋆-open set
of Ycontaining F(x). Then, we have
F(x)∩Cl⋆(Y−Cl⋆(V)) = ∅
and hence x∈ F−(Cl⋆(Y−Cl⋆(V))). It follows
from (7) that x∈ ⋆θsCl(F−(Y−Cl⋆(V))). Then,
there exists a semi-I⋆-open set Uof Xcontaining x
such that sClI⋆(U)∩F−(Y−Cl⋆(V)) = ∅; hence
F(sClI⋆(U)) ⊆Cl⋆(V). This shows that Fis upper
θ(⋆)-quasi continuous.
An ideal topological space (X, τ, I)is said to be
I⋆-compact [8] if every cover of Xby ⋆-open sets
of Xhas a finite subcover. A subset Kof an ideal
topological space (X, τ, I)is said to be I⋆-compact
[8] if every cover of Kby ⋆-open sets has a finite sub-
cover.
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Definition 7. An ideal topological space (X, τ, I)is
called quasi H⋆-closed (resp. s⋆-closed) if, for every
⋆-open (resp. semi-I⋆-open) cover {Vα|α∈ ∇} of
X, there exists a finite subset ∇0of ∇such that
X=∪{Cl⋆(Vα)|α∈ ∇0}
(resp. X=∪{sClI⋆(Vα)|α∈ ∇0}).
Theorem 8. Let F: (X, τ, I)→(Y, σ, J)be
a surjective multifunction and F(x)be J⋆-compact
for each x∈X. If Fis upper θ(⋆)-quasi continuous
and (X, τ, I)is s⋆-closed, then (Y, σ, J)is quasi
H⋆-closed.
Proof. Let {Vα|α∈ ∇} be any ⋆-open cover of Y
and let x∈X. Since F(x)is J⋆-compact, there
exists a finite subset ∇(x)of ∇such that
F(x)⊆ ∪{Vα|α∈ ∇(x)}.
Put V(x) = ∪{Vα|α∈ ∇(x)}, then F(x)⊆V(x)
and V(x)is ⋆-open in Y. Since Fis upper θ(⋆)-quasi
continuous, there exists a semi-I⋆-open set U(x)of
Xcontaining xsuch that
F(sClI⋆(U(x))) ⊆Cl⋆(V(x)).
The family {U(x)|x∈X}is a semi-I⋆-open cover
of X. Since (X, τ, I)is s⋆-closed, there exists a fi-
nite number of points, say, x1, x2, ..., xn∈Xsuch
that X=∪{sClI⋆(U(xi)) |i= 1,2, ..., n}. Since
Fis surjective, we obtain
Y=F(X) = n
∪
i=1F(sClI⋆(U(xi)))
⊆n
∪
i=1Cl⋆(V(xi))
=n
∪
i=1 ∪α∈∇(xi)Cl⋆(Vα).
This shows that (Y, σ, J)is quasi H⋆-closed.
Lemma 9. If F: (X, τ, I)→(Y, σ, J)is lower
θ(⋆)-quasi continuous, then for each x∈Xand each
subset Bof Ysuch that F(x)∩⋆θInt(B)=∅, there
exists a semi-I⋆-open set Uof Xcontaining xsuch
that sClI⋆(U)⊆F−(B).
Proof. Suppose that F(x)∩⋆θInt(B)=∅, there ex-
ists a ⋆-open set Vof Ysuch that V⊆Cl⋆(V)⊆B
and F(x)∩V=∅. Since Fis lower θ(⋆)-quasi
continuous, there exists a semi-I⋆-open set Uof X
containing xsuch that F(z)∩Cl⋆(V)=∅for every
z∈sClI⋆(U)and hence sClI⋆(U)⊆F−(B).
The following theorem gives some characteriza-
tions of lower θ(⋆)-quasi continuous multifunctions.
Theorem 10. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis lower θ(⋆)-quasi continuous;
(2) ⋆θsCl(F+(B)) ⊆F+(⋆θCl(B)) for every sub-
set Bof Y;
(3) ⋆θsCl(F+(V)) ⊆F+(Cl⋆(V)) for every ⋆-open
set Vof Y;
(4) F−(V)⊆⋆θsInt(F−(Cl⋆(V))) for every ⋆-
open set Vof Y;
(5) F(⋆θsCl(A)) ⊆⋆θCl(F(A)) for every subset A
of X;
(6) ⋆θsCl(F+(Int⋆(⋆θCl(B)))) ⊆F+(⋆θCl(B))
for every subset Bof Y;
(7) ⋆θsCl(F+(Int⋆(Cl⋆(V)))) ⊆F+(Cl⋆(V)) for
every ⋆-open set Vof Y;
(8) ⋆θsCl(F+(Int⋆(K))) ⊆F+(K)for every R-
J⋆- closed set Kof Y;
(9) ⋆θsCl(F+(Int⋆(K))) ⊆F+(K)for every ⋆-
closed set Kof Y.
Proof. (1) ⇒(2): Let Bbe any subset of Y. Suppose
that x∈ F+(⋆θCl(B)). Then, we have
x∈F−(Y−⋆θCl(B)) = F−(⋆θInt(Y−B)).
Since Fis lower θ(⋆)-continuous, by Lemma 9, there
exists a semi-I⋆-open set Uof Xcontaining xsuch
that sClI⋆(U)⊆F−(Y−B) = X−F+(B). Thus,
sClI⋆(U)∩F+(B) = ∅and hence
x∈ ⋆θsCl(F+(B)).
(2) ⇒(3): This is obvious sine Cl⋆(V) =
⋆θCl(V)for every ⋆-open set Vof Y.
(3) ⇒(4): Let Vbe any ⋆-open set of Y. By (3),
X−⋆θsInt(F−(Cl⋆(V)))
=⋆θsCl(X−F−(Cl⋆(V)))
=⋆θsCl(F+(Y−Cl⋆(V)))
⊆F+(Cl⋆(Y−Cl⋆(V)))
⊆F+(Cl⋆(Y−V))
=F+(Y−V)
=X−F−(V).
Thus, F−(V)⊆⋆θsInt(F−(Cl⋆(V))).
(4) ⇒(1): Let x∈Xand Vbe any ⋆-open set
such that F(x)∩V=∅. Then, we have
x∈F−(V)⊆⋆θsInt(F−(Cl⋆(V))).
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Therefore, there exists a semi-I⋆-open set Uof X
containing xsuch that sClI⋆(U)⊆F−(Cl⋆(V));
hence F(z)∩Cl⋆(V)=∅for every z∈sClI⋆(U).
This shows that Fis lower θ(⋆)-quasi continuous.
(2) ⇒(5): Let Abe any subset of X. Replacing
Bin (2) by F(A), we have
⋆θsCl(A)⊆⋆θsCl(F+(F(A)))
⊆F+(⋆θCl(F(A)))
and hence F(⋆θsCl(A)) ⊆⋆θCl(F(A)).
(5) ⇒(2): Let Bbe any subset of
Y. Replacing Ain (5) by F+(B), we have
F(⋆θsCl(F+(B))) ⊆⋆θCl(F(F+(B))) ⊆⋆θCl(B).
Thus, ⋆θsCl(F+(B)) ⊆F+(⋆θCl(B)).
(3) ⇒(6): Let Bbe any subset of Y. Put V=
Int⋆(⋆θCl(B)) in (3). Then, since ⋆θCl(B)is ⋆-closed
in Y, we have
⋆θsCl(F+(Int⋆(⋆θCl(B))))
⊆F+(Cl⋆(Int⋆(⋆θCl(B))))
⊆F+(⋆θCl(B)).
(6) ⇒(7): This is obvious since Cl⋆(V) =
⋆θCl(V)for every ⋆-open set Vof Y.
(7) ⇒(8): Let Kbe any R-J⋆-closed set of Y.
By (7), we have
⋆θsCl(F+(Int⋆(K)))
=⋆θsCl(F+(Int⋆(Cl⋆(Int⋆(K)))))
⊆F+(Cl⋆(Int⋆(K)))
=F+(K).
(8) ⇒(9): Let Kbe any ⋆-closed set of Y. Since
Cl⋆(Int⋆(K)) is R-J⋆-closed in Yand by (8),
⋆θsCl(F+(Int⋆(K)))
=⋆θsCl(F+(Int⋆(Cl⋆(Int⋆(K)))))
⊆F+(Cl⋆(Int⋆(K)))
⊆F+(K).
(9) ⇒(4): Let Vbe any ⋆-open set of Y. Then,
Y−Vis ⋆-closed in Yand by (9),
⋆θsCl(F+(Int⋆(Y−V))) ⊆F+(Y−V)
=X−F−(V).
Moreover, we have
⋆θsCl(F+(Int⋆(Y−V)))
=⋆θsCl(F+(Y−Cl⋆(V)))
=⋆θsCl(X−F−(Cl⋆(V)))
=X−⋆θsInt(F−(Cl⋆(V)))
and hence F−(V)⊆⋆θsInt(F−(Cl⋆(V))).
Definition 11. A function (X, τ, I)→(Y, σ, J)is
said to be θ(⋆)-quasi continuous if, for each x∈X
and each ⋆-open set Vof Ycontaining f(x), there
exists a semi-I⋆-open set Uof Xcontaining xsuch
that f(⋆θsCl(U)) ⊆Cl⋆(V).
Corollary 12. For a function f: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) fis θ(⋆)-quasi continuous;
(2) ⋆θsCl(f−1(B)) ⊆f−1(⋆θCl(B)) for every sub-
set Bof Y;
(3) ⋆θsCl(f−1(V)) ⊆f−1(Cl⋆(V)) for every ⋆-
open set Vof Y;
(4) f−1(V)⊆⋆θsInt(f−1(Cl⋆(V))) for every ⋆-
open set Vof Y;
(5) f(⋆θsCl(A)) ⊆⋆θCl(f(A)) for every subset A
of X.
For a multifunction F: (X, τ, I)→(Y, σ, J),
a multifunction sClI⋆F: (X, τ, I)→(Y, σ, J)
is defined as follows: (sClI⋆F)(x) = sClJ⋆(F(x))
for each x∈X.
Lemma 13. Let F: (X, τ, I)→(Y, σ, J)be a
multifunction. Then, (sClI⋆F)−(V) = F−(V)for
each semi-J⋆-open set Vof Y.
Proof. Suppose that Vis any semi-J⋆-open set of
Y. Let x∈(sClI⋆F)−(V). Then,
sClJ⋆(F(x)) ∩V=∅
and so F(x)∩V=∅. Thus, x∈F−(V)and hence
(sClI⋆F)−(V)⊆F−(V). On the other had, let
x∈F−(V). Then, we have ∅ =F(x)∩V⊆
sClJ⋆(F(x)) ∩Vand hence x∈(sClI⋆F)−(V).
This shows that F−(V)⊆(sClI⋆F)−(V). Conse-
quently, we obtain (sClI⋆F)−(V) = F−(V).
Theorem 14. A multifunction F: (X, τ, I)→
(Y, σ, J)is lower θ(⋆)-quasi continuous if and only
if sClI⋆F: (X, τ, I)→(Y, σ, J)is lower θ(⋆)-
quasi continuous.
Proof. Suppose that Fis lower θ(⋆)-quasi continu-
ous. Let x∈Xand let Vbe any ⋆-open sets of Ysuch
that (sClI⋆F)(x)∩V=∅. By Lemma 13, we have
F(x)∩V=∅. Sine Fis lower θ(⋆)-quasi continuous,
there exists a semi-I⋆-open set Uof Xcontaining x
such that F(z)∩Cl⋆(V)=∅for every z∈sClI⋆(U).
Since Cl⋆(V)is semi-J⋆-open in Y, by Lemma 13,
sClI⋆(U)⊆F−(Cl⋆(V)) = (sClI⋆F)−(Cl⋆(V))
and hence (sClI⋆F)(z)∩Cl⋆(V)=∅for every
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z∈sClI⋆(U). This shows that sClI⋆Fis lower
θ(⋆)-quasi continuous.
Conversely, suppose that sClI⋆Fis lower θ(⋆)-
quasi continuous. Let x∈Xand let Vbe any
⋆-open set of Ysuch that F(x)∩V=∅. Then,
sClI⋆(F(x)) ∩V=∅. There exists a semi-I⋆-open
set Uof Xcontaining xsuch that
(sClI⋆F)(z)∩Cl⋆(V)=∅
for every z∈sClI⋆(U). Since Cl⋆(V)is semi-J⋆-
open in Y, by Lemma 13, we have
sClI⋆(U)⊆(sClI⋆F)−(Cl⋆(V))
=F−(Cl⋆(V))
and so F(z)∩Cl⋆(V)=∅for every z∈sClI⋆(U).
This shows that Fis lower θ(⋆)-quasi continuous.
Theorem 15. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis upper θ(⋆)-quasi continuous;
(2) ⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)) for
every semi-J⋆-preopen set Vof Y;
(3) ⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)) for
every semi-J⋆-open set Vof Y.
Proof. (1) ⇒(2): Let Vbe any semi-J⋆-preopen
set of Y. Then, we have V⊆Cl⋆(Int⋆(Cl⋆(V)))
and hence Cl⋆(V) = Cl⋆(Int⋆(Cl⋆(V))). Since
Cl⋆(V)is R-J⋆-closed, by Theorem 6, we have
⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)).
(2) ⇒(3): This is obvious since every semi-J⋆-
open set is semi-J⋆-preopen.
(3) ⇒(1): Let Vbe any ⋆-open set of Y. By
(3),⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)) and
by Theorem 6, Fis upper θ(⋆)-quasi continuous.
Theorem 16. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis lower θ(⋆)-quasi continuous;
(2) ⋆θsCl(F+(Int⋆(Cl⋆(V)))) ⊆F+(Cl⋆(V)) for
every semi-J⋆-preopen set Vof Y;
(3) ⋆θsCl(F+(Int⋆(Cl⋆(V)))) ⊆F+(Cl⋆(V)) for
every semi-J⋆-open set Vof Y.
Proof. The proof is similar to that of Theorem 15.
Corollary 17. For a function f: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) fis θ(⋆)-quasi continuous;
(2) ⋆θsCl(f−1(Int⋆(Cl⋆(V)))) ⊆f−1(Cl⋆(V)) for
every semi-J⋆-preopen set Vof Y;
(3) ⋆θsCl(f−1(Int⋆(Cl⋆(V)))) ⊆f−1(Cl⋆(V)) for
every semi-J⋆-open set Vof Y.
Theorem 18. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis upper θ(⋆)-quasi continuous;
(2) ⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(V)) for
every J⋆-preopen set Vof Y;
(3) ⋆θsCl(F−(V)) ⊆F−(Cl⋆(V)) for every J⋆-
preopen set Vof Y;
(4) F+(V)⊆⋆θsInt(F+(Cl⋆(V))) for every J⋆-
preopen set Vof Y.
Proof. (1) ⇒(2): Let Vbe any J⋆-preopen set of
Y. Since Int⋆(Cl⋆(V)) is ⋆-open in Y, by Theorem 6,
⋆θsCl(F−(Int⋆(Cl⋆(V)))) ⊆F−(Cl⋆(Int⋆(Cl⋆(V))))
=F−(Cl⋆(V)).
(2) ⇒(3): Let Vbe any J⋆-preopen set of Y.
Then, we have V⊆Int⋆(Cl⋆(V)) and by (2),
⋆θsCl(F−(V)) ⊆⋆θsCl(F−(Int⋆(Cl⋆(V))))
⊆F−(Cl⋆(V)).
(3) ⇒(4): Let Vbe any J⋆-preopen set of Y.
By (3), we have
X−⋆θsInt(F+(Cl⋆(V)))
=⋆θsCl(X−F+(Cl⋆(V)))
=⋆θsCl(F−(Y−Cl⋆(V)))
⊆F−(Cl⋆(Y−Cl⋆(V)))
=X−F+(Int⋆(Cl⋆(V)))
⊆X−F+(V)
and hence F+(V)⊆⋆θsInt(F+(Cl⋆(V))).
(4) ⇒(1): Let Vbe any ⋆-open set of Y. Then,
we have Vis J⋆-preopen and by (4),
F+(V)⊆⋆θsInt(F+(Cl⋆(V))).
Thus, Fis upper θ(⋆)-quasi continuous by Theorem
6.
Theorem 19. For a multifunction F: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) Fis lower θ(⋆)-quasi continuous;
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(2) ⋆θsCl(F+(Int⋆(Cl⋆(V)))) ⊆F+(Cl⋆(V)) for
every J⋆-preopen set Vof Y;
(3) ⋆θsCl(F+(V)) ⊆F+(Cl⋆(V)) for every J⋆-
preopen set Vof Y;
(4) F−(V)⊆⋆θsInt(F−(Cl⋆(V))) for every J⋆-
preopen set Vof Y.
Proof. The proof is similar to that of Theorem 18.
Corollary 20. For a function f: (X, τ, I)→
(Y, σ, J), the following properties are equivalent:
(1) fis lower θ(⋆)-quasi continuous;
(2) ⋆θsCl(f−1(Int⋆(Cl⋆(V)))) ⊆f−1(Cl⋆(V)) for
every J⋆-preopen set Vof Y;
(3) ⋆θsCl(f−1(V)) ⊆f−1(Cl⋆(V)) for every J⋆-
preopen set Vof Y;
(4) f−1(V)⊆⋆θsInt(f−1(Cl⋆(V))) for every J⋆-
preopen set Vof Y.
4 Conclusion
Topology as a field of mathematics is concerned with
all questions directly or indirectly related to continu-
ity. Generalization of continuity is one of the main re-
search topics in general topology. This paper is deal-
ing with the concepts of upper and lower θ(⋆)-quasi
continuous multifunctions. Moreover, some charac-
terizations of upper and lower θ(⋆)-quasi continuous
multifunctions are obtained. The ideas and results of
this paper may motivate further research.
Acknowledgements
This research project was financially supported by
Mahasarakham University.
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7
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