EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 430-439
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Forms of Open Multifunctions in Ideal
Topological Spaces

Takashi Noiri 1,∗, Valeriu Popa2

1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan
2 Department of Mathematics, University of Vasile Alecsandri of Bacǎu, 600115 Bacǎu,
Romania

Abstract. By using m-open multifunctions from an m-space into an m-space, we establish the
unified theory for several weak forms of open multifunctions between topological spaces.

2020 Mathematics Subject Classifications: 54A05, 54C10, 54C60

Key Words and Phrases: m-structure, m-space, m-open multifunction, ideal

1. Introduction

The notion of ideal topological spaces is introduced in [15] and [27]. In [14], the authors
introduced the notion of I-open sets in an ideal topological space. As generalizations of
open sets and I-open sets, semi-I-open sets, pre-I-open sets, α-I-open sets, β-I-open sets
and b-I-open sets are introduced and used to obtain decompositions of continuity.

Recently, in [24] and [25] the present authors introduced the notions of minimal struc-
tures and m-spaces as a generalization of topological spaces. The notion of m-open mul-
tifunctions is introduced in [21]. The notion of m-I-open functions is introduced in [22].
In this paper, the authors introduce a minimal structure mIO(X) determined by opera-
tions Int, Cl, Cl⋆ in an ideal topological space (X, τ, I). By using mIO(X), the authors
introduce and study the notion of mI-open multifunctions. As special case of mI-open
multifunctions, we obtain semi-I-open functions [12], pre-I-open functions [2], α-I-open
functions [2], b-I-open functions [3], weakly semi-I-open functions [9] and weakly b − I-
open functions [19].

In Section 3, we introduce the notion of an m-open multifunction from an m-space
into an m-space. We obtain the characterizations of m-open multifunctions and charac-
terize the set of all points at which a multifunction is not m-open. In the last part, a new
modification of m-open multifunctions, called mI-open multifunctions, is introduced and
investigated.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4691

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

https://www.ejpam.com 430 © 2023 EJPAM All rights reserved.



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 431

2. Preliminaries

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

Definition 1. Let (X, τ) be a topological space. A subset A of X is said to be α-open [20]
(resp. semi-open [16], preopen [18], β-open [1], b-open [4]) if A ⊂ Int(Cl(Int(A))) (resp.
A ⊂ Cl(Int(A)), A ⊂ Int(Cl(A)), A ⊂ Cl(Int(Cl(A))), A ⊂ Int(Cl(A)) ∪ Cl(Int(A))).

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

Throughout the present paper, (X, τ) and (Y, σ) always denote topological spaces and
F : X → Y presents a multivalued function. For a multifunction F : X → Y , we shall
denote the upper and lower inverse of a subset B of a space 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 ̸= ∅}.

Definition 2. A multifunction F : (X, τ) → (Y, σ) is said to be open [5] (resp. semi-
open [23], preopen [7], α-open [6], β-open [21]) if F (U) is open (resp. semi-open, preopen,
α-open, β-open) for each open set U of X.

Definition 3. A subfamily mX of the power set P(X) of a nonempty set X is called a
minimal structure (or briefly m-structure) [24], [25] on X if ∅ ∈ mX and X ∈ mX .

By (X,mX) (or briefly (X,m)), we denote a nonempty set X with a minimal structure
mX on X and call it an m-space. Each member of mX is said to be mX -open (or briefly
m-open) and the complement of an mX -open set is said to be mX-closed (or briefly m-
closed).

Definition 4. Let X be a nonempty set and mX an m-structure on X. For a subset A
of X, the mX-closure and the mX-interior of A are defined in [17] as follows:

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

Lemma 1. (Maki et al. [17]) Let (X,mX) be an m-space. For subsets A and B of X, the
following properties hold:

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

Definition 5. An m-structure mX on a nonempty set X is said to have property B [17]
if the union of any family of subsets belonging to mX belongs to mX .



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 432

Remark 1. Let (X, τ) be a topological space and mX = SO(X) (resp. PO(X), α(X),
β(X), BO(X)), then mX is an m-structure having property B.

Lemma 2. (Popa and Noiri [26]) Let (X,mX) be an m-space and mX have property B.
Then for a subset A of X, the following properties hold:

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

3. m-open multifunctions

Definition 6. Let (X,mX) and (Y,mY ) be twom-spaces. A multifunction F : (X,mX) →
(Y,mY ) is said to be m-open at x ∈ X if for each mX -open set U containing x, there exists
V ∈ mY containing F (x) such that V ⊂ F (U). If F is m-open at each point x ∈ X, then
F is said to be m-open.

Theorem 1. A multifunction F : (X,mX) → (Y,mY ) is m-open at x ∈ X, where mY has
property B, if and only if for each mX-open set U containing x, x ∈ F+(mY Int(F (U))).

Proof. Necessity. Let U be any mX -open set containing x. Then, there exists V ∈ mY

such that F (x) ⊂ V ⊂ F (U) and hence F (x) ⊂ mY Int(F (U)). Therefore, we obtain that
x ∈ F+(mY Int(F (U))).

Sufficiency. Suppose that x ∈ F+(mY Int(F (U))) for each mX -open set U containing
x. Then F (x) ⊂ mY Int(F (U)). Set V = mY Int(F (U)), then by Lemma 2 V ∈ mY and
F (x) ⊂ V ⊂ F (U). Therefore, F is m-open at x.

Theorem 2. A multifunction F : (X,mX) → (Y,mY ) is m-open, where (Y,mY ) has
property B, if and only if F (U) is mY -open for each mX-open set U of X.

Proof. Necessity. Let U be any mX -open set of X and x ∈ U . Since F is m-open
at x ∈ X, by Theorem 1 we have F (x) ⊂ mY Int(F (U)) and F (U) = mY Int(F (U)). By
Lemma 2, F (U) is mY -open.

Sufficiency. Let x be an arbitrary point of X and U any mX -open set of X containing
x. Then, we have F (x) ⊂ F (U) = mY Int(F (U)). Therefore, x ∈ F+(mY Int(F (U)). By
Theorem 1, F is m-open at an arbitrary point x ∈ X.

Remark 2. For a multifunction F : (X,mX) → (Y,mY ), let mX = τ and mY = σ (resp.
SO(Y ), PO(Y ), α(Y ), β(Y )), then we obtain Definition 2, that is, the definition of an
open (resp. semi-open, preopen, α-open, β-open) multifunction.

Theorem 3. For a multifunction F : (X,mX) → (Y,mY ), where mY has property B, the
following properties are equivalent:

(1) F is m-open at x;
(2) If x ∈ mXInt(A) for any A ∈ P(X), then x ∈ F+(mY Int(F (A)));
(3) If x ∈ mXInt(F+(B)) for any B ∈ P(Y ), then x ∈ F+(mY Int(B));
(4) If x ∈ F−(mY Cl(B)) for any B ∈ P(Y ), then x ∈ mXCl(F−(B)).



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 433

Proof. (1) ⇒ (2): Let A ∈ P(X) and x ∈ mXInt(A). Then, there exists an mX -
open set U such that x ∈ U ⊂ A and hence F (x) ⊂ F (U) ⊂ F (A). Since F is m-open
at x, by Theorem 1 and Lemma 1, we obtain x ∈ F+(mY Int(F (U))) ⊂ F+(mY Int(F (A))).

(2)⇒ (3): LetB ∈ P(Y ) and x ∈ mXInt(F+(B)). Then, x ∈ F+(mY Int(F (F+(B)))) ⊂
F+(mY Int(B)).

(3) ⇒ (4): Let B ∈ P(Y ) and x /∈ mXCl(F−(B)). Then x ∈ X − mXCl(F−(B)) =
mXInt(X − F−(B)) = mXInt(F+(Y − B)). By (3) we have x ∈ F+(mY Int(Y − B)) =
X − F−(mY Cl(B)). Hence, x /∈ F−(mY Cl(B)). Therefore, if x ∈ F−(mY Cl(B)), then
x ∈ mXCl(F−(B)).

(4) ⇒ (1): Let U be any mX -open set of X containing x and B = Y − F (U). Since
mXCl(F−(B)) = mXCl(F−(Y − F (U))) = mXCl(X − F+(F (U))) ⊂ X − mXInt(U) =
X−U and x ∈ U , we obtain that x /∈ mXCl(F−(B)). By (4), we have x /∈ F−(mY Cl(B)) =
F−(mY Cl(Y − F (U))) = X − F+(mY Int(F (U))). Therefore, x ∈ F+(mY Int(F (U))). By
Theorem 1, F is m-open at x.

Theorem 4. For a multifunction F : (X,mX) → (Y,mY ), where mY has property B, the
following properties are equivalent:

(1) F is m-open;
(2) F (mXInt(A)) ⊂ mY Int(F (A)) for any subset A of X;
(3) mXInt(F+(B)) ⊂ F+(mY Int(B)) for any subset B of Y;
(4) F−(mY Cl(B)) ⊂ mXCl(F−(B)) for any subset B of Y.

Proof. (1) ⇒ (2): Let A be any subset of X and x ∈ mXInt(A). Since F is m-open
at each x ∈ A, by Theorem 3 F (x) ⊂ mY Int(F (A)). Hence F (mXInt(A)) ⊂ mY Int(F (A)).

(2)⇒ (3): LetB be any subset of Y . By (2), we have F (mXInt(F+(B))) ⊂ mY Int(F (F+(B))) ⊂
mY Int(B). Hence, we have mXInt(F+(B)) ⊂ F+(mY Int(B)) .

(3) ⇒ (4): Let B be any subset of Y . By (3), we have X − mXCl(F−(B)) =
mXInt(X − F−(B)) = mXInt(F+(Y −B)) ⊂ F+(mY Int(Y −B)) = X − F−(mY Cl(B)).
Hence, F−(mY Cl(B)) ⊂ mXCl(F−(B)).

(4) ⇒ (1): Let U be any mX -open set of X and B = Y − F (U). By (4), we
have F−(mY Cl(Y − F (U))) ⊂ mXCl(F−(Y − F (U))). Now, F−(mY Cl(Y − F (U))) =
F−(Y − mY Int(F (U))) = X − F+(mY Int(F (U))). And also we have mXCl(F−(Y −
F (U))) = mXCl(X − F+(F (U))) ⊂ X − mXInt(U) = X − U . Therefore, we obtain
U ⊂ F+(mY Int(F (U))) and hence F (U) ⊂ mY Int(F (U)). Consequently, we obtain
F (U) = mY Int(F (U)) and F (U) is mY -open. Therefore, by Theorem 2 F is m-open.

For a multifunction F : (X,mX) → (Y,mY ), we denote

D0(F ) = {x ∈ X: F is not m-open at x }.

Theorem 5. For a multifunction F : (X,mX) → (Y,mY ), where mY has property B, the
following properties hold:

D0(F ) = ∪U∈mX
{U − F+(mY Int(F (U)))}

= ∪A∈P (X){mXInt(A)− F+(mY Int(F (A)))}



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 434

= ∪B∈P (Y ){mXInt(F+(B))− F+(mY Int(B))}
= ∪B∈P (Y ){F−(mY Cl(B))−mXCl(F−(B))}.

Proof. Let x ∈ D0(F ). Then, by Theorem 1, there exists anmX -open set U0 containing
x such that x /∈ F+(mY Int(F (U0))). Hence, x ∈ U0 ∩ (X − F+(mY Int(F (U0)))) =
U0 − F+(mY Int(F (U0))) ⊂ ∪U∈mX

{U − F+(mY Int(F (U)))}.
Conversely, let x ∈ ∪U∈mX

{U−F+(mY Int(F (U)))}. Then, there exists U0 ∈ mX such
that x ∈ U0 − F+(mY Int(F (U0))). Therefore, by Theorem 1 x ∈ D0(F ).

For the second equation, let x ∈ D0(F ). Then, by Theorem 3, there exists A1 ∈ P
(X) such that x ∈ mXInt(A1) and x /∈ F+(mY Int(F (A1))). Therefore, x ∈ mXInt(A1)−
F+(mY Int(F (A1))) ⊂ ∪A∈P (X){mXInt(A)− F+(mY Int(F (A)))}.

Conversely, x ∈ ∪A∈P (X){mXInt(A) − F+(mY Int(F (A)))}. Then, there exists A1 ∈
P(X) such that x ∈ mXInt(A1)− F+(mY Int(F (A1))). By Theorem 3, x ∈ D0(F ).

The other equations are similarly proved.

4. Ideal topological spaces

Let (X, τ) be a topological space. The notion of ideals has been introduced in [15] and
[27] and further investigated in [13]

Definition 7. 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 [13]. 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⋆.

Lemma 3. (Janković and Hamlett [13]) Let (X, τ, I) be an ideal topological space and A,
B be two 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).

A subset A is said to be I-open [14] if A ⊂ Int(A⋆). As generalizations of open sets
and I-open sets, the following subsets are introduced and investigated.

Definition 8. Let (X, τ, I) be an ideal topological space. A subset A of X is said to be
(1) α-I-open [11] if A ⊂ Int(Cl⋆(Int(A))),
(2) semi-I-open [12] if A ⊂ Cl⋆(Int(A)),
(3) pre-I-open [8] if A ⊂ Int(Cl⋆(A)),



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 435

(4) b-I-open [3] if A ⊂ Int(Cl⋆(A)) ∪ Cl⋆(Int(A)),
(5) β-I-open [11] if A ⊂ Cl(Int(Cl⋆(A))),
(6) weakly semi-I-open [9] if A ⊂ Cl⋆(Int(Cl(A))),
(7) weakly b-I-open [19] if A ⊂ Cl(Int(Cl⋆(A))) ∪ Cl⋆(Int(Cl(A))),
(8) strongly β-I-open [10] if A ⊂ Cl⋆(Int(Cl⋆(A))).

Between the sets in Definition 8, we have the following relations:

DIAGRAM 1

open ⇒ α-I-open ⇒ semi-I-open ⇒ weakly semi-I-open
⇓ ⇓ ⇓

I-open ⇒ pre-I-open ⇒ b-I-open ⇒ weakly b-I-open
⇓ ⇑

strongly β-I-open ⇒ β-I-open

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) 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)).

Definition 9. 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).

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

Proof. We shall show that SIO(X) is an m-structure with property B.
(1) It is obvious that by Lemma 3 Cl⋆(Int(∅)) = Cl⋆(∅) = ∅ and Cl⋆(Int(X)) =

Cl⋆(X) = X. Hence, SIO(X) is an m-structure.
(2) Let {Aα : α ∈ ∆} be any family of semi-I-open sets. Then, for each α ∈ ∆, by

Lemma 3 we have Aα ⊂ Cl⋆(Int(Aα)) ⊂ Cl⋆(Int(∪{Aα : α ∈ ∆})). Therefore, ∪{Aα : α ∈
∆} ⊂ Cl⋆(Int(∪{Aα : α ∈ ∆})) and hence ∪{Aα : α ∈ ∆} ∈ SIO(X). Hence SIO(X) has
property B.
For other families, the proofs are similar.

Definition 10. 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 m-structure on X. If
mIO(X) = τ⋆ (resp. αIO(X), SIO(X), PIO(X), BIO(X), βIO(X), WSIO(X), WBIO(X),
SβIO(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)).

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



T. Noiri, V. Popa / Eur. J. Pure Appl. Math, 16 (1) (2023), 430-439 436

5. mI-open multifunctions

Definition 11. Let (X,mX) be an m-space and (Y, τ, I) be an ideal topological space. A
multifunction F : (X,mX) → (Y, τ, I) is said to be mI-open at x ∈ X if for each mX -open
set U containing x, there exists V ∈ mIO(Y ) containing F (x) such that V ⊂ F (U). If F
is mI-open at each point x ∈ X, then F is said to be mI-open.

Then F : (X,mX) → (Y, τ, I) is mI-open at x ∈ X (resp. on X) if and only if
F : (X,mX) → (Y,mIO(X)) is m-open at x ∈ X (resp. on X). Therefore, by the results
of Section 3, we obtain the following properties of mI-open multifunctions.

Theorem 6. A multifunction F : (X,mX) → (Y, τ, I) is mI-open at x ∈ X if and only if
for each mX-open set U containing x, x ∈ F+(mIntI(F (U))).

Proof. The proof follows from Theorem 1 and Lemma 4.

Theorem 7. A multifunction F : (X,mX) → (Y, τ, I) is mI-open if and only if F (U) is
mI-open for each mX-open set U of X.

Proof. The proof follows from Theorem 2 and Lemma 4.

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

(1) F is mI-open at x;
(2) If x ∈ mXInt(A) for A ∈ P(X), then x ∈ F+(mIntI(F (A)));
(3) x ∈ mXInt(F+(B)) for B ∈ P(Y ), then x ∈ F+(mIntI(B));
(4) If x ∈ F−(mClI(B)) for B ∈ P(Y ), then x ∈ mXCl(F−(B)).

Proof. The proof follows from Theorem 3 and Lemma 4.

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

(1) F is mI-open;
(2) F (mXInt(A)) ⊂ mIntI(F (A)) for any subset A of X;
(3) mXInt(F+(B)) ⊂ F+(mIntI(B)) for any subset B of Y;
(4) F−(mClI(B)) ⊂ mXCl(F−(B)) for any subset B of Y.

Proof. The proof follows from Theorem 4 and Lemma 4.

For a multifunction F : (X,mX) → (Y, τ, I), we denote

D0
I (F ) = {x ∈ X: F is not mI-open at x }.

Theorem 10. For a multifunction F : (X,mX) → (Y, τ, I), the following properties hold:
D0

I (F ) = ∪U∈mX
{U − F−(mIntI(F (U)))}

= ∪A∈P (X){mXInt(A)− F+(mIntI(F (A)))}
= ∪B∈P (Y ){mXInt(F+(B))− F+(mIntI(B))}
= ∪B∈P (Y ){F−(mClI(B))−mXCl(F−(B))}.



REFERENCES 437

Proof. The proof follows from Theorem 5 and Lemma 4.

Remark 3. 1) Let F : (X, τ) → (Y, σ, J) be a multifunction, where (X, τ) is a topolog-
ical space. Since mX = SO(X) (resp. PO(X), α(X), β(X), BO(X)) is an m-structure
having property B, an mJ-open multifunction F : (X,mX) → (Y, σ, J) is defined and it is
equivalent to an m-open multifunction F : (X,mX) → (Y,mJO(Y )). For example, let
mX = SO(X) and mJO(Y ) = SJO(Y ), then an m-open multifunction F : (X,SO(X)) →
(Y,SJO(Y )) is defined and we obtain the properties from the results of Sections 3 and 5.

2) An mIJ-open multifunction F : (X, τ, I) → (Y, σ, J) is defined by (i) an mJ-
open multifunction F : (X,mIO(X)) → (Y, σ, J) or (ii) an m-open multifunction F :
(X,mIO(X)) → (Y,mJO(Y )).

For example, let mIO(X) = SIO(X) and mJO(Y ) = SJO(Y ), then an m-open multi-
function F : (X,SIO(X)) → (Y, SJO(Y )) is defined and we obtain the properties from the
results of Sections 3 and 5.

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

(1) F : (X, τ, I) → (Y, σ, J) is mIJ-open;
(2) F : (X,SIO(X)) → (Y, σ, J) is mJ-open;
(3) F : (X,SIO(X)) → (Y, SJO(Y )) is m-open;
(4) F (sIntI(A)) ⊂ sIntJ(F (A)) for any subset A of X;
(5) sIntI(F

+(B)) ⊂ F+(sIntJ(B)) for any subset B of Y;
(6) F−(sClJ(B)) ⊂ sClI(F

−(B)) for any subset B of Y.

Proof. The proof easily follows from Theorem 9.

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