EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 1-14
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On m-I-Continuous Multifunctions

Takashi Noiri1,∗, Valeriu Popa2

1 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan
2 Department of Mathematics, University Vasile Alecsandri of Bacau, 600 115-Bacau,
Romania

Abstract. Let mIO(X) be the family of ⋆-open (resp. α-I-open, pre-I-open, semi-I-open, β-I-
open, etc.) sets in an ideal topological space (X, τ, I). By using mIO(X), we introduce and inves-
tigate the notions of an m-I-continuous multifunction F : (X, τ, I) → (Y, σ) and mi⋆-continuous
multifunction F : (X, τ, I) → (Y, σ, J) . The notion of mi⋆-continuity is a generalization of m-I-
continuity and i⋆-continuity [9].

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Minimal structure, ideal topological space, m-I-open, m-I-continuous,
mi⋆-continuous, multifunction

1. Introduction

Semi-open sets, pre-open sets, α-open sets, b-open sets and β-open sets play an impor-
tant role in the research of generalizations of continuity for functions and multifunctions. In
1961, Marcus [23] introduced the notion of quasicontinuity in topological spaces. Neubrun-
nova [26] showed that quasicontinuity is equivalent to semi-continuity due to Levine [21].
Bânzaru [6] and Bânzaru and Crivǎţ [7] extended it to the notion of quasicontinuity for
multifunctions. Properties of quasicontinuous multifunctions are further investigated in
[13], [33], and [39].

The present authors introduced and studied α-continuous multifunctions [36], pre-
continuous multifunctions [39], β-continuous multifunctions [37]. Przemski [46] also intro-
duced the notions of α-continuity, precontinuity and presemi-continuity for multifunctions.
It is poved in [36] (resp. [39], [37]) that the notion of α-continuity (resp. precontinuity,
β-continuity) for multifunctions in the sense of Popa and Noiri is equivalent to that of
α-continuity (resp. precontinuity, presemi-continuity) in the sense of Przemski.

The notions of minimal structure, m-continuity, M -continuity are introduced in [40]
and [41]. By using these notions, the present authors unified theory of continuity in [42],

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4207

Email addresses: t.noiri@nifty.com (T. Noiri), vpopa@ub.ro (V. Popa)

http://www.ejpam.com 1 © 2022 EJPAM All rights reserved.



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 2

[44], and [28] and other papers. The upper/lower m-continuous (resp. M -continuous)
multifunctions are introduced and investigated in [42], [44] (resp. [28], [29]) and other
papers.

The notion of ideal topological spaces was introduced in [20], [47]. As generelariza-
tions of open sets, the notions of semi-I-open sets, pre-I-open sets, α-I-open sets, b-I-open
sets and β-I-open sets are inroduced and studied. The notion of upper/lower-Icontinuous
multifunctions is introduced in [2]. Quite recently other results are obtained in [8], [9], [4],
[31] and other papers.

In this paper, by mIO(X) we denote the family of ⋆-open (resp. semi-I-open, pre-
I-open, α-I-open, b-I-open, β-I-open, etc.) sets in an ideal topological space (X, τ, I).
Then we introduce and investigate the notion of an m-I-continuous multifunction F :
(X, τ, I) → (Y, σ) which generalizes the results obtained in [36], [37] and [39]. Further-
more, we introduce the notion of an mi⋆-continuous multifunction F : (X, τ, I) → (Y, σ, J)
which generalizes the notions of i⋆-continuous multifunctions [9] and m-I-continuous mul-
tifunctions.

2. Preliminaries

Let (X, τ) be a topological spacce and A a subset of X. The closure of A and the
interior of A are denoted by Cl(A) and Int(A), respectively.

Definition 1. A subset A of a topological space (X, τ) is said to be
(1) α-open [27] if A ⊂ Int(Cl(Int(A))),
(2) semi-open [21] if A ⊂ Cl(Int(A)),
(3) preopen [24] if A ⊂ Int(Cl(A)),
(4) b-open [3] if A ⊂ Cl(Int(A)) ∪ Int(Cl(A)),
(5) β-open [1] if A ⊂ Cl(Int(Cl(A))).

The family of all semi-open (resp. preopen, α-open, b-open, β-open) sets in (X, τ) is
denoted by SO(X) (resp. PO(X), α(X), BO(X), β(X)).

Throughout the present paper, spaces (X, τ) and (Y, σ) always mean topological spaces
and F : (X, τ) → (Y, σ) presents a multivalued function. For a multifunction, we shall
denote the upper and lower inverses of a subset B of Y by F+(B) and F−(B), respectively,
that is,

F+(B) = {x ∈ X : F (x) ⊂ B} and F−(B) = {x ∈ X : F (x) ∩B ̸= ∅}.
Let P(Y ) be the collection of all nonempty subsets of Y . For any open set V of Y , we

denote V + = {B ∈ P(Y ) : B ⊂ V } and V − = {B ∈ P(Y ) : B ∩ V ̸= ∅} [46].

Definition 2. A multifunction F : (X, τ) → (Y, σ) is said to be quasi-continuous [6], [7],
[33] (resp. precontinuous [39], α-continuous [36], β-continuous [37]) at a point x ∈ X if
for each open sets G1, G2 of Y such that F (x) ∈ G+

1 ∩G−
2 , there exists a semi-open (resp.

preopen, α-open, β-open) set U of X containing x such that F (u) ∈ G+
1 ∩ G−

2 for every
u ∈ U . A multifunction is said to be quasi-continuous (resp. precontinuous, α-continuous,
β-continuous) if it has this property at each point of x ∈ X.



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 3

3. m-continuous multifunctions

Definition 3. A subfamily mX of the power set P(X) of a nonempty set X is called a
minimal structure (briefly m-structure) on X if ∅ ∈ mX and X ∈ mX . Each member of
mX is said to be mX-open (briefly m-open) and the complement of an mX -open set is
said to be mX-closed. (briefly m-closed). A set X with an mX -structure mX is called an
m-space and is denoted by (X,mX)

Remark 1. Let (X, τ) be a topological space. Then the families τ , α(X), SO(X), PO(X),
BO(X), β(X) are all m-structures on X.

Definition 4. Let X be a nonempty set and mX an m-structure on X. For a subset A
of X, the mX-closure of A and the mX-interior of A are defined in [22] as follows:

(1) mCl(A) = ∩{F : A ⊂ F,X − F ∈ mX},
(2) mInt(A) = ∪{U : U ⊂ A,U ∈ mX}.

Remark 2. Let (X, τ) be a topological space and A a subset of X. If mX = τ (resp.
SO(X), PO(X), BO(X), α(X), β(X)), then we have

(1) mCl(A) = Cl(A) (resp. sCl(A), pCl(A), bCl(A), αCl(A), βCl(A)),
(2) mInt(A) = Int(A) (resp. sInt(A), pInt(A), bInt(A), αInt(A), βInt(A)).

Lemma 1. ([22]). Let (X,mX) be an m-space. For subsets A and B of X, the following
properties hold:

(1) mCl(X −A) = X −mInt(A) and mInt(X −A) = X −mCl(A),
(2) If (X −A) ∈ mX , then mCl(A) = A and if A ∈ mX , then mInt(A) = A,
(3) mCl(∅) = ∅, mCl(X) = X, mInt(∅) = ∅ and mInt(X) = X,
(4) If A ⊂ B, then mCl(A) ⊂ mCl(B) and mInt(A) ⊂ mInt(B),
(5) A ⊂ mCl(A) and mInt(A) ⊂ A,
(6) mCl(mCl(A)) = mCl(A) and mInt(mInt(A)) = mInt(A).

Definition 5. A minimal structure mX on a nonempty set X is said to have property B
[22] if the union of any family of subsets belonging to mX belongs to mX .

Remark 3. Let (X, τ) be a topological space. Then the families τ , SO(X), PO(X), α(X),
BO(X) and β(X) are all minimal structures having property B.

Lemma 2. Let X be a nonempty set and mX an m-structure with property B. Then, the
following properties are hold:

(1) mInt(A) = A if and only if A ∈ mX ,
(2) mCl(A) = A if and only if A is m-closed,
(3) mInt(A) ∈ mX and mCl(A) is m-closed.

Definition 6. A multifunction F : (X,mX) → (Y, σ) is said to be m-continuous at x ∈ X
[42] if for each open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩ V −
2 , there exists U ∈ mX

containing x such that F (u) ∈ V +
1 ∩ V −

2 for every u ∈ U . F is said to be m-continuous if
it has the property at each point of X.



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 4

Remark 4. Let F : (X,mX) → (Y, σ) be a multifunction. If mX = SO(X) (resp. PO(X),
α(X), BO(X), β(X)), then F is quasi-continuous (resp. precontinuous, α-continuous, b-
continuous, β-continuous).

Theorem 1. ([44]). For a multifunction F : (X,mX) → (Y, σ), the following properties
are equivalent:

(1) F is m-continuous at x ∈ X;
(2) F (x) ∈ V +

1 ∩ V −
2 implies x ∈ mInt[F+(V1)∩F−(V2)] for every open sets V1, V2 of

Y;
(3) x ∈ mCl(F−(B1) ∪ F+(B2)) implies x ∈ F−(Cl(B1)) ∪ F+(Cl(B2)) for every

subsets B1, B2 of Y;
(4) x ∈ F−(Int(B1)) ∩ F+(Int(B2)) implies x ∈ Int(F−(B1) ∩ F+(B2)) for every

subsets B1, B2 of Y.

Theorem 2. ([42]). For a multifunction F : (X,mX) → (Y, σ), the following properties
are equivalent:

(1) F is m-continuous;
(2) F+(G1) ∩ F−(G2) = mInt(F+(G1) ∩ F−(G2)) for every open sets G1, G2 of Y ;
(3) F−(K1) ∪ F+(K2) = mCl(F−(K1) ∪ F+(K2)) for every closed sets K1,K2 of Y;
(4) mCl(F−(B1) ∪ F+(B2)) ⊂ F−(Cl(B1)) ∪ F+(Cl(B2)) for every subsets B1, B2 of

Y;
(5) F−(Int(B1))∩F+(Int(B2)) ⊂ mInt(F−(B1)∩F+(B2)) for every subsets B1, B2 of

Y .

For a multifunction F : (X,mX) → (Y, σ), we define Dm(F ) as follows:
Dm(F ) = {x ∈ X : F is not m-continuous at x}.

Theorem 3. ([44]). For a multifunction F : (X,mX) → (Y, σ), the following equalities
hold:

Dm(F ) =
⋃

G1,G2∈σ{F
+(G1) ∩ F−(G2)−mInt(F+(G1) ∩ F−(G2))}

=
⋃

B1,B2∈P (Y ){F−(Int(B1)) ∩ F+(Int(B2))−mInt(F−(B1) ∩ F+(B2))}
=

⋃
B1,B2∈P (Y ){mCl(F−(B1) ∪ F+(B2))− [F−(Cl(B1)) ∪ F+(Cl(B2))]}

=
⋃

H1,H2∈F{mCl(F−(H1) ∪ F+(H2))− [F−(H1) ∪ F+(H2)]},
where F is the family of closed sets of (Y, σ).

Definition 7. ([42]). Let (X,mX) be an m-space. For a subset A of X, the mX-frontier
mFr(A) of A is defined as follows:

mFr(A) = mCl(A) ∩mCl(X −A).

Theorem 4. ([42]). The set of all points x ∈ X at which a multifunction F : (X, τ, I) →
(Y, σ) is not m-continuous is identical with the union of the mX-frontiers of the intersec-
tions of upper/lower inverse images of open sets containing/meeting F (x).

Definition 8. A subset B of a topological space (Y, σ) is said to be
(1) α-regular [19] if for each b ∈ B and any open set U containing b, there exists an



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 5

open set G of Y such that b ∈ G ⊂ Cl(G) ⊂ U ,
(2) α-paracompact [48] if every σ-open cover of B has a σ-open refinement which covers

B and is locally finite for each point of Y .

For a multifunction F : (X,mX) → (Y, σ), by Cl(F ) : X → Y [6] we denote a
multifunction defined as follows: Cl(F )(x) = Cl(F (x)) for each x ∈ X. Similarly, sCl(F )
(resp. pCl(F ), αCl(F ), bCl(F ), βCl(F ) ) is defined in [32] (resp. [34], [35], [8], [38]).

Theorem 5. ([42]). Let F : (X,mX) → (Y, σ) be a multifunction such that F (x) is α-
regular and α-paracompact for each x ∈ X. Then the following properties are equivalent:

(1) F is m-continuous;
(2) G is m-continuous, where G = Cl(F ), sCl(F ), pCl(F ), αCl(F ), bCl(F ), and

βCl(F ).

Definition 9. ([42]). A multifunction F : (X,mX) → (Y, σ) is said to be
(1) upper m-continuous at x ∈ X if for each open set V containing F (x), there exists

U ∈ mX containing x such that F (U) ⊂ V ,
(2) lower m-continuous at x ∈ X if for each open set V meeting F (x), there exists

U ∈ mX containing x such that F (u) ∩ V ̸= ∅ for every u ∈ U ,
(3) upper/lower m-continuous if it has this property at each point x ∈ X.

Theorem 6. ([42]). Let X be a nonempty set with two m-structures m1
X and m2

X satisfying
property B such that V1 ∈ m1

X and V2 ∈ m2
X implies V1 ∩ V2 ∈ m1

X . If a multifunction
F : (X,m1

X) → (Y, σ) is upper m-continuous and F : (X,m2
X) → (Y, σ) is lower m-

continuous, then F : (X,m1
X) → (Y, σ) is m-continuous.

Theorem 7. ([42]). Let X be a nonempty set with two m-structures m1
X and m2

X satisfying
property B such that V1 ∈ m1

X and V2 ∈ m2
X implies V1 ∩ V2 ∈ m1

X . If a multifunction
F : (X,m1

X) → (Y, σ) is lower m-continuous and F : (X,m2
X) → (Y, σ) is upper m-

continuous, then F : (X,m1
X) → (Y, σ) is m-continuous.

4. Ideal topological spaces

Let (X, τ) be a topological space. The notion of ideals has been introduced in [20] and
[47] and further investigated in [18]

Definition 10. A nonempty collection I of subsets of a set X is called an ideal on X if
it satisfies the following two conditions:

(1) A ∈ I and B ⊂ A implies B ∈ I,
(2) A ∈ I and B ∈ I implies A ∪B ∈ I.

A topological space (X, τ) with an ideal I on X is called an ideal topological space and
is denoted by (X, τ, I). Let (X, τ, I) be an ideal topological space. For any subset A of
X, A⋆(I, τ) = {x ∈ X : U ∩ A /∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U},
is called the local function of A with respect to τ and I [18]. Hereafter A⋆(I, τ) is simply
denoted by A⋆. It is well known that Cl⋆(A) = A ∪ A⋆ defines a Kuratowski closure
operator on X and the topology generated by Cl⋆ is denoted by τ⋆.



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 6

Lemma 3. Let (X, τ, I) be an ideal topological space and A, B be subsets of X. Then the
following properties hold:

(1) A ⊂ B implies Cl⋆(A) ⊂ Cl⋆(B),
(2) Cl⋆(X) = X and Cl⋆(∅) = ∅,
(3) Cl⋆(A) ∪ Cl⋆(B) ⊂ Cl⋆(A ∪B).

Definition 11. Let (X, τ, I) be an ideal topological space. A subset A of X is said to be
(1) α-I-open [16] if A ⊂ Int(Cl⋆(Int(A))),
(2) semi-I-open [16] if A ⊂ Cl⋆(Int(A)),
(3) pre-I-open [10] if A ⊂ Int(Cl⋆(A)),
(4) b-I-open [5] if A ⊂ Int(Cl⋆(A)) ∪ Cl⋆(Int(A)),
(5) β-I-open [17] if A ⊂ Cl(Int(Cl⋆(A))),
(6) weakly semi-I-open [14] if A ⊂ Cl⋆(Int(Cl(A))),
(7) weakly b-I-open [25] if A ⊂ Cl(Int(Cl⋆(A))) ∪ Cl⋆(Int(Cl(A))),
(8) strongly β-I-open [15] if A ⊂ Cl⋆(Int(Cl⋆(A))),
(9) semi⋆-I-open [12] if A ⊂ Cl(Int⋆(A)),
(10) pre⋆-I-open [11] if A ⊂ Int⋆(Cl(A)),
(11) β⋆

I -open [11] if A ⊂ Cl(Int⋆(Cl(A))).

The family of all α-I-open (resp. semi-I-open, pre-I-open, b-I-open, β-I-open, weakly
semi-I-open, weakly b-I-open, strongly β-I-open, semi⋆-I-open, pre⋆-I-open, β⋆

I -open)
sets in an ideal topological space (X, τ, I) is denoted by αIO(X) (resp. SIO(X), PIO(X),
BIO(X), βIO(X), WSIO(X), WBIO(X), SβIO(X), S⋆IO(X), P⋆IO(X), βIO(X)).

Definition 12. By mIO(X), we denote each one of the families τ⋆, αIO(X), SIO(X),
PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X), SβIO(X), S⋆IO(X), P⋆IO(X), β⋆IO(X).

Lemma 4. Let (X, τ, I) be an ideal topological space. Then mIO(X) is a minimal structure
and has property B.

Definition 13. Let (X, τ, I) be an ideal topological space. For a subset A of X, mClI(A)
and mIntI(A) are defined as follows:

(1) mClI(A) = ∩{F : A ⊂ F,X \ F ∈ mIO(X)},
(2) mIntI(A) = ∪{U : U ⊂ A,U ∈ mIO(X)}.

Let (X, τ, I) be an ideal topological space and mIO(X) the mX -structure on X. If
mIO(X) = τ⋆ (resp. αIO(X), SIO(X), PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X),
SβIO(X)), S⋆IO(X), P⋆IO(X), β⋆IO(X), then we have the following:

(1) mClI(A) = Cl⋆(A) (resp. αClI(A), sClI(A), pClI(A), bClI(A), βClI(A), wsClI(A),
wbClI(A), sβClI(A), s⋆ClI(A), p⋆ClI(A), β⋆ClI(A)),

(2) mIntI(A) = Int⋆(A) (resp. αIntI(A), sIntI(A), pIntI(A), bIntI(A), βIntI(A), wsIntI(A),
wbIntI(A), sβIntI(A), s⋆IntI(A), p⋆IntI(A), β⋆IntI(A)).

5. m-I-continuous multifunctions

Definition 14. A multifunction F : (X, τ, I) → (Y, σ) is said to be m-I-continuous at
x ∈ X if for each open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩V −
2 , there exists U ∈ mIO(X)



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 7

containing x such that F (u) ∈ V +
1 ∩ V −

2 for every u ∈ U . F is said to be m-I-continuous
if it has the property at each point of X.

By Theorem 1 and Definition 13, we obtain the following theorem.

Theorem 8. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are
equivalent:

(1) F is m-I-continuous at x ∈ X;
(2) F (x) ∈ V +

1 ∩ V −
2 implies x ∈ mIntI[F

+(V1)∩F−(V2)] for every open sets V1, V2 of
Y;

(3) x ∈ mClI(F
−(B1) ∪ F+(B2)) implies x ∈ F−(Cl(B1)) ∪ F+(Cl(B2)) for every

subsets B1, B2 of Y;
(4) x ∈ F−(Int(B1)) ∩ F+(Int(B2)) implies x ∈ mIntI(F

−(B1) ∩ F+(B2)) for every
subsets B1, B2 of Y.

By Theorem 2 and Definition 13, we obtain the following theorem:

Theorem 9. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are
equivalent:

(1) F is m-I-continuous;
(2) F+(G1) ∩ F−(G2) ∈ mIO(X) for every open sets G1, G2 of Y ;
(3) F−(K1) ∪ F+(K2) is m-I-closed for every closed sets K1,K2 of Y;
(4) mClI(F

−(B1) ∪ F+(B2)) ⊂ F−(Cl(B1)) ∪ F+(Cl(B2)) for every subsets B1, B2 of
Y;

(5) F−(Int(B1)) ∩ F+(Int(B2)) ⊂ mIntI(F
−(B1) ∩ F+(B2)) for every subsets B1, B2

of Y .

Let mIO(X) = τ⋆, then by Theorem 9, we obtain the following corollary:

Corollary 1. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are
equivalent:

(1) F is τ⋆-continuous;
(2) F+(G1) ∩ F−(G2) ∈ τ⋆ for every open sets G1, G2 of Y ;
(3) F−(K1) ∪ F+(K2) is τ⋆-closed for every closed sets K1,K2 of Y;
(4) Cl⋆(F−(B1)∪F+(B2)) ⊂ F−(Cl(B1))∪F+(Cl(B2)) for every subsets B1, B2 of Y;
(5) F−(Int(B1)) ∩ F+(Int(B2)) ⊂ Int⋆(F−(B1) ∩ F+(B2)) for every subsets B1, B2 of

Y .

Let mIO(X) = SIO(X), then by Theorem 9, we obtain the following corollary:

Corollary 2. For a multifunction F : (X, τ, I) → (Y, σ), the following properties are
equivalent:

(1) F is semi-I-continuous;
(2) F+(G1) ∩ F−(G2) ∈ SIO(X) for every open sets G1, G2 of Y ;
(3) F−(K1) ∪ F+(K2) is semi-I-closed for every closed sets K1,K2 of Y;
(4) sClI(F

−(B1) ∪ F+(B2)) ⊂ F−(Cl(B1)) ∪ F+(Cl(B2)) for every subsets B1, B2 of



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 8

Y;
(5) F−(Int(B1))∩F+(Int(B2)) ⊂ sIntI(F

−(B1)∩F+(B2)) for every subsets B1, B2 of
Y .

For a multifunction F : (X, τ, I) → (Y, σ), we define DmI(F ) as follows:

DmI(F ) = {x ∈ X : F is not m-I-continuous at x}.

Theorem 10. For a multifunction F : (X, τ, I) → (Y, σ), the following equalities hold:
Dm(F ) =

⋃
G1,G2∈σ{F

+(G1) ∩ F−(G2)−mIntI(F
+(G1) ∩ F−(G2))]}

=
⋃

B1,B2∈P (Y ){F−(Int(B1)) ∩ F+(Int(B2))−mIntI(F
−(B1) ∩ F+(B2))}

=
⋃

B1,B2∈P (Y ){mClI(F
−(B1) ∪ F+(B2))− [F−(Cl(B1)) ∪ F+(Cl(B2))]}

=
⋃

H1,H2∈F{mClI(F
−(H1) ∪ F+(H2))− [F−(H1) ∪ F+(H2)]},

where F is the family of closed sets of (Y, σ).

Let mIO(X) = SIO(X), then by Theorem 10 we obtain the following corollary.

Corollary 3. For a multifunction F : (X, τ, I) → (Y, σ), the following equalities hold:
Dm(F ) =

⋃
G1,G2∈σ{F

+(G1) ∩ F−(G2)− sIntI(F
+(G1) ∩ F−(G2))]}

=
⋃

B1,B2∈P (Y ){F−(Int(B1)) ∩ F+(Int(B2))− sIntI(F
−(B1) ∩ F+(B2))}

=
⋃

B1,B2∈P (Y ){sClI(F−(B1) ∪ F+(B2))− [F−(Cl(B1)) ∪ F+(Cl(B2))]}
=

⋃
H1,H2∈F{sClI(F−(H1) ∪ F+(H2))− [F−(H1) ∪ F+(H2)]},

where F is the family of closed sets of (Y, σ).

Definition 15. Let (X, τ, I) be an ideal topological space. For a subset A of X, the
mI-frontier mIFr(A) of A is defined as follows:

mIFr(A) = mClI(A) ∩mClI(X −A).

Theorem 11. The set of all points x ∈ X at which a multifunction F : (X, τ, I) → (Y, σ)
is not m-I-continuous is identical with the union of the mI-frontiers of the intersection of
upper/lower inverse images of open sets containing/meeting F (x).

Proof. The proof follows from Definition 13 and Theorem 4.

If mIO(X) = τ⋆, then we obtain the following corollary:

Corollary 4. The set of all points x ∈ X at which a multifunction F : (X, τ, I) → (Y, σ)
is not τ⋆-continuous is identical with the union of the τ⋆-frontiers of the intersection of
upper/lower inverse images of open sets containing/meeting F (x).

If mIO(X) = SIO(X), then we obtain the following corollary:

Corollary 5. The set of all points x ∈ X at which a multifunction F : (X, τ, I) → (Y, σ)
is not semi-I-continuous is identical with the union of the SI-frontiers of the intersection
of upper/lower inverse images of open sets containing/meeting F (x).



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 9

Theorem 12. Let F : (X, τ, I) → (Y, σ) be a multifunction such that F (x) is α-regular
and α-paracompact for each x ∈ X. Then the following properties are equivalent:

(1) F is m-I-continuous;
(2) G is m-I-continuous, where G = Cl(F (x)), sCl(F ), pCl(F ), αCl(F ), bCl(F),

βCl(F ).

Proof. The proof follows from Theorem 5.

Definition 16. A multifunction F : (X, τ, I) → (Y, σ) is said to be
(1) upper m-I-continuous at x ∈ X if for each open set V containing F (x), there exists

U ∈ mIO(X) containing x such that F (U) ⊂ V ,
(2) lower m-I-continuous at xinX if for each open set V meeting F (x), there exists

U ∈ mIO(X) containing x such that F (u) ∩ V ̸= ∅ for every u ∈ U ,
(3) upper/lower m-I-continuous if it has this property at each point x ∈ X.

Theorem 13. Let X be a nonempty set with two m-structures m1
X and m2

X satisfying
property B such that V1 ∈ m1

X and V2 ∈ m2
X implies V1 ∩ V2 ∈ m1

X . If a multifunction
F : (X, τ, I) → (Y, σ) is upper m1

X-I-continuous and F : (X, τ, I) → (Y, σ) is lower
m2

X-I-continuous, then F : (X, τ, I) → (Y, σ) is m1
X-I-continuous.

Theorem 14. Let X be a nonempty set with two m-structures m1
X and m2

X satisfying
property B such that V1 ∈ m1

X and V2 ∈ m2
X implies V1 ∩ V2 ∈ m1

X . If a multifunction
F : (X, τ, I) → (Y, σ) is lower m1

X-continuous and F : (X, τ, I) → (Y, σ) is upper m2
X-

continuous, then F : (X, τ, I) → (Y, σ) is m1
X-I-continuous.

6. mi⋆-continuous multifunctions

A multifunction F : (X, τ, I) → (Y, σ, J) is said to be i⋆-continuous [9] if for each x ∈ X
and each σ⋆-open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩ V −
2 , there exists a τ⋆-open set

U containing x such that F (U) ⊂ V +
1 and F (u) ∩ V2 ̸= ∅ for every u ∈ U .

Definition 17. A multifunction F : (X, τ, I) → (Y, σ, J) is said to be mi⋆-continuous if
for each x ∈ X and each σ⋆-open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩ V −
2 , there exists

an mIO(X)-open set U containing x such that F (U) ⊂ V +
1 and F (u) ∩ V2 ̸= ∅ for every

u ∈ U .

Remark 5. For a multifunction F : (X, τ, I) → (Y, σ, J), we have the following properties:
(1) If mIO(X) = τ⋆, then every mi⋆-continuous multifunction is i⋆-continuous. There-

fore, the notion of mi⋆-continuity is a generalization of i⋆-continuity.
(2) If J = {∅}, then σ⋆ = σ. Therefore, the notion of mi⋆-continuity is a generalization

of m-I-continuity.

Theorem 15. For a multifunction F : (X, τ, I) → (Y, σ, J), the following properties are
equivalent:

(1) F is mi⋆-continuous;



Takashi Noiri, Valeriu Popa / Eur. J. Pure Appl. Math, 15 (1) (2022), 1-14 10

(2) For each point x ∈ X and each σ⋆-open sets V1, V2 of Y such that F (x) ∈ V +
1 ∩V −

2 ,
x ∈ mIntI(F

+(V1) ∩ F−(V2));
(3) F+(V1) ∩ F−(V2) ∈ mIO(X) for every σ⋆-open sets V1, V2 of Y ;
(4) F−(K1) ∪ F+(K2) is m-I-closed for every σ⋆-closed sets K1,K2 of Y;
(5) mClI(F

−(B1) ∪ F+(B2)) ⊂ F−(Cl⋆(B1)) ∪ F+(Cl⋆(B2)) for every subsets B1, B2

of Y;
(6) F−(Int⋆(B1))∩F+(Int⋆(B2)) ⊂ mIntI(F

−(B1)∩F+(B2)) for every subsets B1, B2

of Y .

Proof. (1) => (2): Let x ∈ X and V1, V2 be any σ⋆-open sets of Y such that
F (x) ∈ V +

1 ∩V −
2 . Then there exists U ∈ mIO(X) containing x such that F (U) ∈ V +

1 ∩V −
2 .

Therefore, U ⊂ F+(V1) ∩ F−(V2) and hence x ∈ mIntI(F
+(V1) ∩ F−(V2)).

(2) => (3): Let V1, V2 be any σ⋆-open sets of Y and x ∈ F+(V1) ∩ F−(V2). Then
F (x) ⊂ V1 and F (x) ∩ V2 ̸= ∅. By (2), we have x ∈ mIntI(F

+(V1) ∩ F−(V2)) and
F+(V1)∩F−(V2) ⊂ mIntI(F

+(V1)∩F−(V2)). This shows that F
+(V1)∩F−(V2) ∈ mIO(X).

(3) => (4): This easily follows from the fact that F−(Y − B) = X − F+(B) and
F+(Y −B) = X − F−(B) for every subset B of Y .

(4) => (5): B1, B2 be any subsets of Y . Then Cl⋆(B1) and Cl⋆(B2) are σ⋆-closed.
By (4), mClI(F

−(B1)∪ F+(B2)) ⊂ mClI(F
−(Cl⋆(B1))∪ F+(Cl⋆(B2))) = (F−(Cl⋆(B1))∪

F+(Cl⋆(B2)).
(5) => (6): B1, B2 be any subsets of Y . By (5), we have

X−mIntI(F
−(B1)∩F+(B2)) = mClI(X− (F−(B1)∩F+(B2))) = mClI((X−F−(B1))∪

(X−F+(B2))) = mClI(F
+(Y −B1)∪F−(Y −B2)) ⊂ F+(Cl⋆(Y −B1))∪F−(Cl⋆(Y −B2)) =

(X − F−(Int⋆(B1))) ∪ (X − F+(Int⋆(B2))) = X − (F−(Int⋆(B1)) ∩ F+(Int⋆(B2))).

Therefore, we obtain F−(Int⋆(B1)) ∩ F+(Int⋆(B2)) ⊂ mIntI(F
−(B1) ∩ F+(B2)).

(6) => (1): Let x ∈ X and V1, V2 be any σ⋆-open sets of Y such that F (x) ∈ V +
1 ∩V −

2 .
By (6), F−(V1)∩F+(V2) ⊂ mIntI(F

−(V1)∩F+(V2)). This shows that F
−(V1)∩F+(V2) ∈

mIO(X). And put U = F−(V1) ∩ F+(V2). Then U is an mIO(X)-open set containing x
such that F (U) ⊂ V +

1 and F (u)∩V −
2 ̸= ∅ for every u ∈ U . Therefore, F is mi⋆-continuous.

If mIO(X) = SIO(X), by Theorem 15 we obtain the following corollary:

Corollary 6. For a multifunction F : (X, τ, I) → (Y, σ, J), the following properties are
equivalent:

(1) F is si⋆-continuous;
(2) For each point x ∈ X and each σ⋆-open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩V −
2 ,

x ∈ sIntI(F
+(V1) ∩ F−(V2));

(3) F+(V1) ∩ F−(V2) ∈ SIO(X) for every σ⋆-open sets V1, V2 of Y ;
(4) F−(K1) ∪ F+(K2) is semi-I-closed for every σ⋆-closed sets K1,K2 of Y;
(5) sClI(F

−(B1)∪F+(B2)) ⊂ F−(Cl⋆(B1))∪F+(Cl⋆(B2)) for every subsets B1, B2 of
Y;

(6) F−(Int⋆(B1)) ∩ F+(Int⋆(B2)) ⊂ sIntI(F
−(B1) ∩ F+(B2)) for every subsets B1, B2

of Y .



REFERENCES 11

If mIO(X) = PIO(X), by Theorem 15 we obtain the following corollary:

Corollary 7. For a multifunction F : (X, τ, I) → (Y, σ, J), the following properties are
equivalent:

(1) F is pi⋆-continuous;
(2) For each point x ∈ X and each σ⋆-open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩V −
2 ,

x ∈ pIntI(F
+(V1) ∩ F−(V2));

(3) F+(V1) ∩ F−(V2) ∈ PIO(X) for every σ⋆-open sets V1, V2 of Y ;
(4) F−(K1) ∪ F+(K2) is pre-I-closed for every σ⋆-closed sets K1,K2 of Y;
(5) pClI(F

−(B1)∪F+(B2)) ⊂ F−(Cl⋆(B1))∪F+(Cl⋆(B2)) for every subsets B1, B2 of
Y;

(6) F−(Int⋆(B1))∩ F+(Int⋆(B2)) ⊂ pIntI(F
−(B1)∩ F+(B2)) for every subsets B1, B2

of Y .

Theorem 16. The set of all points x ∈ X at which a multifunction F : (X, τ, I) →
(Y, σ, J) is not mi⋆-continuous is identical with the union of the mI-frontiers of the inter-
section of upper/lower inverse images of ⋆-open sets containing/meeting F (x).

Proof. The proof follows similarly from Theorem 11.

If mIO(X) = τ⋆, then we obtain the following corollary:

Corollary 8. The set of all points x ∈ X at which a multifunction F : (X, τ, I) → (Y, σ, J)
is not τ⋆-continuous is identical with the union of the τ⋆-frontiers of the intersection of
upper/lower inverse images of ⋆-open sets containing/meeting F (x).

If mIO(X) = SIO(X), then we obtain the following corollary:

Corollary 9. The set of all points x ∈ X at which a multifunction F : (X, τ, I) → (Y, σ, J)
is not si⋆-continuous is identical with the union of the SI-frontiers of the intersection of
upper/lower inverse images of ⋆-open sets containing/meeting F (x).

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