# On Characterizations of ∗-Hyperconnected Ideal Topological Spaces

Authors: Chawalit Boonpok
Journal: Journal of Mathematics
 (2020)

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## Abstract

In the present paper, some characterizations of∗-hyperconnected ideal topological spaces are investigated. Moreover, we introduce the notion of S∗-ℐ-hyperconnected ideal topological spaces. Several characterizations of S∗-ℐ-hyperconnected ideal topological spaces are discussed. Furthermore, we introduce and study θ-ℐ-irreducible ideal topological spaces.


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## Body

## 1. Introduction

The concept of hyperconnected spaces was introduced by Steen and Seebach [1]. Several concepts which are equivalent to hyperconnectedness were defined and investigated in the literature. Levine [2] called a topological space X a D-space if every nonempty open set of X is dense in X and showed that X is a D-space if and only if it is hyperconnected. Pipitone and Russo [3] defined a topological space X to be semiconnected if X is not the union of two disjoint nonempty semiopen sets of X and showed that X is semiconnected if and only if it is a D-space. Sharma [4] indicated that a space is a D-space if it is a hyperconnected space. Ajmal and Kohli [5] have investigated further properties of hyperconnected spaces. In [6], the author obtained several characterizations of hyperconnected spaces by using semi-preopen sets and almost feebly continuous functions. Hyperconnected spaces are also called irreducible in [7]. D-spaces, s-connected spaces [8], semiconnected spaces, and irreducible spaces are equivalent to hyperconnected spaces. Janković and Long [9] have introduced and investigated the notion of θ-irreducible spaces. It is pointed out in [9] that hyperconnectedness implies θ-irreducibility and θ-irreducibility implies connectedness. Noiri and Umehara [10] investigated some characterizations and some preservation theorems concerning θ-irreducible spaces. The concept of ideal topological spaces was introduced and studied by Kuratowski [11] and Vaidyanathaswamy [12]. Janković and Hamlett [13] investigated further properties of ideal topological spaces. Hatir and Noiri [14] introduced the notions of semi-ℐ-open sets, α-ℐ-open sets, and β-ℐ-open sets in topological spaces via ideals and used these sets to obtain certain decomposition of continuity. Later, in [15], the same authors investigated further properties of semi-ℐ-open sets and semi-ℐ-continuous functions introduced in [14]. Recently, Ekici and Noiri [16] have introduced the notion of ∗-hyperconnected ideal topological spaces and investigated some characterizations of ∗-hyperconnected ideal topological spaces. The purpose of the present paper is to investigate some characterizations of ∗-hyperconnected ideal topological spaces. Moreover, we introduce and study the notion of S∗-ℐ-hyperconnected ideal topological spaces. In the last section, we introduce the notion of θ-ℐ-irreducible ideal topological spaces. Especially, several characterizations of θ-ℐ-irreducible ideal topological spaces are established.


## 2. Preliminaries

Throughout the present paper, spacesX,τ and Y,σ (or simply X and Y) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. In a topological space X,τ, the closure and the interior of any subset A of X will be denoted by ClA and IntA, respectively. An ideal ℐ on a topological space X,τ is a nonempty collection of subsets of X satisfying the following properties: (1) A∈ℐ and B⊆A imply B∈ℐ; (2) A∈ℐ and B∈ℐ imply 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. For an ideal topological space X,τ,ℐ and a subset A of X, A∗ℐ is defined as follows: A∗ℐ=x∈X:U∩A∉ℐ for every open neighbourhood U of x. In case there is no chance for confusion, A∗ℐ is simply written as A∗. In [11], A∗ is called the local function of A with respect to ℐ and τ and Cl∗A=A∗∪A defines a Kuratowski closure operator for a topology τ∗ℐ. For every ideal topological space X,τ,ℐ, there exists a topology τ∗ℐ finer than τ, generated by the base ℬℐ,τ=U−IU∈τandI∈ℐ. However, ℬℐ,τ is not always a topology [12]. A subset A is said to be ∗-closed [13] if A∗⊆A. The interior of a subset A in X,τ∗ℐ is denoted by Int∗A.Definition 1.
A subsetA of an ideal topological space X,τ,ℐ is said to be(1)
Pre-ℐ-open [17] if A⊆IntCl∗A(2)
Semi-ℐ-open [14] if A⊆Cl∗IntA(3)
Semi∗-ℐ-open [16] if A⊆ClInt∗A(4)
Strongβ-ℐ-open [18] if A⊆Cl∗IntCl∗A(5)
R-ℐ-open [19] if A=IntCl∗A(6)
b-ℐ-open [20] if A=IntCl∗A∪Cl∗IntA(7)
∗-Dense [21] if Cl∗A=X
BypℐOX,τ (resp. sℐOX,τ, s∗ℐOX,τ, sβℐOX,τ, rℐOX,τ, and bℐOX,τ), we denote the family of all pre-ℐ-open (resp. semi-ℐ-open, semi∗-ℐ-open, strong β-ℐ-open, R-ℐ-open, and b-ℐ-open) sets of an ideal topological space X,τ,ℐ.
The complement of a pre-ℐ-open (resp. semi-ℐ-open, semi∗-ℐ-open, strong β-ℐ-open, and R-ℐ-open) set is called pre-ℐ-closed [17] (resp. semi-ℐ-closed [14], semi∗-ℐ-closed [16, 22], strong β-ℐ-closed [18], and R-ℐ-closed [19]).Definition 2 (see [16]).
The pre-ℐ-closure (resp. semi-ℐ-closure, semi∗-ℐ-closure, and strong β-ℐ-closure) of a subset A of an ideal topological space X,τ,ℐ, denoted by pClℐA (resp. sClℐA, s∗ClℐA, and sβClℐA), is defined by the intersection of all pre-ℐ-closed (resp. semi-ℐ-closed, semi∗-ℐ-closed, and strong β-ℐ-closed) sets of X containing A.Lemma 1 (see [16]).
For a subsetA of an ideal topological space X,τ,ℐ, the following properties hold:(1)
pClℐA=A∪ClInt∗A(2)
sClℐA=A∪Int∗ClA(3)
s∗ClℐA=A∪IntCl∗A(4)
sβClℐA=A∪Int∗ClInt∗ALemma 2 (see [23]).
LetA be a subset of an ideal topological space X,τ,ℐ and U be an open set. Then, U∩Cl∗A⊆Cl∗U∩A.


## 3. On Characterizations of∗-Hyperconnected Ideal Topological Spaces

First, we investigate some characterizations of ∗-hyperconnected ideal topological spaces defined by Ekici and Noiri [16].Definition 3 (see [16]).
An ideal topological spaceX,τ,ℐ is said to be ∗-hyperconnected if V is ∗-dense for every nonempty open set V of X.
The following theorems give several characterizations of ∗-hyperconnected ideal topological spaces.Theorem 1.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is ∗-hyperconnected(2)
Cl∗U=X for every nonempty set U∈pℐOX,τ(3)
s∗ClℐU=X for every nonempty set U∈pℐOX,τProof.
1⟹2: suppose that X,τ,ℐ is ∗-hyperconnected. Let U be a nonempty pre-ℐ-open set. Then, U is strong β-ℐ-open; by Theorem 14 of [16], Cl∗U=X.2⟹3: let U be a nonempty pre-ℐ-open set. By (2), Cl∗U=X, and hence X=IntCl∗U=U∪IntCl∗U=s∗ClℐU.3⟹1: let U be a nonempty open set. Then, U is pre-ℐ-open, and by (3), Cl∗U⊇s∗ClℐU=X. Thus, X,τ,ℐ is ∗-hyperconnected.Lemma 3.
LetX,τ,ℐ be an ideal topological space. Then, s∗ClℐV∈sℐOX,τ for every V∈sℐOX,τ.Proof.
It follows from Lemma 1(3) and Theorem 3.4(1) of [15].Theorem 2.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is ∗-hyperconnected(2)
U∩V≠∅ for every nonempty set U∈sℐOX,τ and every nonempty set V∈s∗ℐOX,τ(3)
The only subsets ofX which are both semi-ℐ-open and semi∗-ℐ-closed in X,τ,ℐ are empty set and X itself(4)
X cannot be expressed by the disjoint union of a nonempty semi-ℐ-open set and a nonempty semi∗-ℐ-open set(5)
s∗ClℐV=X for every nonempty set V∈sℐOX,τProof.
1⟹2: it follows from Theorem 11 of [16].2⟹3: let V be a subset of X, which is both semi-ℐ-open and semi∗-ℐ-closed. Then, V is semi-ℐ-open and X−V is semi∗-ℐ-open such that V∩X−V=∅, which is a contradiction of (2).3⟹4: suppose that X=U∪V, where U is a nonempty semi-ℐ-open set and V a nonempty semi∗-ℐ-open set such that U∩V=∅. Since X−U=V and V is semi∗-ℐ-open, U is semi∗-ℐ-closed. This shows that U is both semi-ℐ-open and semi∗-ℐ-closed, which is a contradiction of (3).4⟹5: suppose that s∗ClℐV≠X for some nonempty set V∈sℐOX,τ. Then, we have X−s∗ClℐV≠∅, s∗ClℐV≠∅ and(1)X=X−s∗ClℐV∪s∗ClℐV.Sinces∗ClℐV is semi∗-ℐ-closed, X−s∗ClℐV is semi∗-ℐ-open, and by Lemma 3, s∗ClℐV is semi-ℐ-open, which is a contradiction of (4).5⟹1: let V be a nonempty open set. Then, V is semi-ℐ-open, and by (5), X=s∗ClℐV⊆Cl∗V. Thus, Cl∗V=X. This shows that X,τ,ℐ is ∗-hyperconnected.
Next, we will introduce and study the concept ofS∗-ℐ-hyperconnected ideal topological spaces.Definition 4.
An ideal topological spaceX,τ,ℐ is called S∗-ℐ-hyperconnected if s∗ClℐU=X for every nonempty open set U.Remark 1.
For an ideal topological spaceX,τ,ℐ, we have the following diagram:(2)X,τ,ℐisS∗‐ℐ‐hyperconnected⟹X,τ,ℐis∗‐hyperconnected.
The implication in the diagram is not reversible as shown in the following example.Example 1 (see [16]).
LetX=a,b,c,d with a topology(3)τ=∅,X,a,c,a,b,a,c,a,b,c,a,c,d,and an ideal ℐ=∅,b. Then X,τ,ℐ is ∗-hyperconnected, but X,τ,ℐ is not S∗-ℐ-hyperconnected.Lemma 4 (see [15]).
A subsetA of an ideal topological space X,τ,ℐ is semi-ℐ-open if and only if there exists U∈τ such that U⊆A⊆Cl∗U.Theorem 3.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is S∗-ℐ-hyperconnected(2)
s∗ClℐA=X or Ints∗ClℐA=∅ for every subset A of X(3)
U∩V≠∅ for every nonempty open set U and every nonempty semi∗-ℐ-open set V(4)
U∩V≠∅ for every nonempty semi-ℐ-open set U and every nonempty semi∗-ℐ-open set VProof.
1⟹2: let A be a subset of X. Suppose that Ints∗ClℐA≠∅. Since X,τ,ℐ is S∗-ℐ-hyperconnected, X=s∗ClℐInts∗ClℐA⊆s∗ClℐA. Thus, s∗ClℐA=X.2⟹3: suppose that U∩V=∅ for some nonempty open set U and nonempty semi∗-ℐ-open set V. Then, s∗ClℐU∩V=∅, and by (2), s∗ClℐU≠X. Moreover, since U is open, ∅≠U⊆Ints∗ClℐU.3⟹4: suppose that U∩V=∅ for some nonempty semi-ℐ-open set U and nonempty semi∗-ℐ-open set V. Since U is a nonempty semi-ℐ-open set, by Lemma 4, there exists a nonempty open set G such that G⊆U⊆Cl∗G. Then, G is a nonempty semi-ℐ-open set such that G∩V=∅, which is a contradiction of (3).4⟹1: suppose that X,τ,ℐ is not S∗-ℐ-hyperconnected. Therefore, there exists a nonempty open set G of X such that s∗ClℐG≠X, and hence X−s∗ClℐG≠∅. Thus, X−s∗ClℐG is a nonempty semi∗-ℐ-open set and G is a nonempty semi-ℐ-open set such that X−s∗ClℐG∩G=∅. This is a contradiction. Consequently, we obtain X,τ,ℐ is S∗-ℐ-hyperconnected.Theorem 4.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is ∗-hyperconnected(2)
Cl∗U=X for every nonempty set U∈pℐOX,τ(3)
s∗ClℐU=X for every nonempty set U∈pℐOX,τ(4)
X,τ,ℐ is S∗-ℐ-hyperconnectedProof.
1⟹2 and 2⟹3 follow from Theorem 1.3⟹4 and 4⟹1 are obvious.
For a subsetA of an ideal topological space X,τ,ℐ, we denote by τA the relative topology on A and ℐA=A∩II∈ℐ is an ideal on A [24].Lemma 5 (see [24]).
LetX,τ,ℐ be an ideal topological space and B⊆A⊆X. Then, B∗τA,ℐA=B∗τ,ℐ∩A.Lemma 6 (see [23]).
LetX,τ,ℐ be an ideal topological space and B⊆A⊆X. Then, ClA∗B=Cl∗B∩A.Lemma 7.
LetU be an open set of an ideal topological space X,τ,ℐ. If V∈pℐOU,τU,ℐU, then V∈pℐOX,τ.Proof.
Suppose thatU is an open set and V∈pℐOU,τU,ℐU. Since V⊆IntUClU∗V and IntUClU∗V is open in U,τU, there exists an open set W such that U∩W=IntUClU∗V. Since U⊆IntCl∗U and by Lemma 2,(4)V⊆IntCl∗U∩W=IntCl∗U∩W⊆IntCl∗U∩W=IntCl∗IntUClU∗V⊆IntCl∗ClU∗V⊆IntCl∗Cl∗V=IntCl∗V.
Thus,V∈pℐOX,τ.Lemma 8.
LetU be an open set of an ideal topological space X,τ,ℐ. Then, s∗ClℐUV=s∗ClℐV∩U for every V⊆U.Proof.
LetV be a subset of U. Then, we have(5)s∗ClℐUV=V∪IntUClU∗V=V∪IntUCl∗V∩U=V∪IntCl∗V∩U=V∪IntCl∗V∩U=V∪IntCl∗V∩U=s∗ClℐV∩U.Theorem 5.
LetU be an open set of an ideal topological space X,τ,ℐ. If X,τ,ℐ is S∗-ℐ-hyperconnected, then U,τU,ℐU is S∗-ℐU-hyperconnected.Proof.
Suppose thatX,τ,ℐ is S∗-ℐ-hyperconnected. Let V be a nonempty pre-ℐU-open set of U,τU,ℐU. By Lemma 7, V is pre-ℐ-open in X,τ,ℐ. Since X,τ,ℐ is S∗-ℐ-hyperconnected, s∗ClℐV=X, and by Lemma 8, s∗ClℐUV=s∗ClℐV∩U=X∩U=U. Therefore, U,τU,ℐU is S∗-ℐU-hyperconnected.Theorem 6.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is S∗-ℐ-hyperconnected(2)
Cl∗V=X for every nonempty open set VProof.
1⟹2: it follows from Theorem 14 of [16].2⟹1: let V be a nonempty open set. By (2), we have(6)X=Cl∗V=IntCl∗V⊆V∪IntCl∗V=s∗ClℐV,and hence s∗ClℐV=X. This shows that X,τ,ℐ is S∗-ℐ-hyperconnected.Definition 5.
A functionf:X,τ,ℐ⟶Y,σ,J is said to be θ-ℐ-continuous if for each point x∈X and each open set V containing fx, there exists an open set U containing x such that fCl∗U⊆Cl∗V.Theorem 7.
Iff:X,τ,ℐ⟶Y,σ,J is a θ-ℐ-continuous surjection and X,τ,ℐ is S∗-ℐ-hyperconnected, then Y,σ,J is S∗-J-hyperconnected.Proof.
Suppose thatX,τ,ℐ is S∗-ℐ-hyperconnected. Let V be a nonempty open set of Y. Since f is surjective, there exists a point x of X such that fx∈V. Since f is θ-ℐ-continuous, there exists an open set U containing x such that fCl∗U⊆Cl∗V. Since X,τ,ℐ is S∗-ℐ-hyperconnected, by Theorem 6, we have Cl∗U=X, and hence Y=fX=fCl∗U⊆Cl∗V. It follows from Theorem 6 that Y,σ,J is S∗-J-hyperconnected.Theorem 8.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is S∗-ℐ-hyperconnected(2)
V is ∗-dense for every nonempty set V∈sℐOX,τ(3)
s∗ClℐV=X for every nonempty set V∈sℐOX,τProof.
The proof follows from Theorem 14 of [16].Theorem 9.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is S∗-ℐ-hyperconnected(2)
V is ∗-dense for every nonempty set V∈sℐOX,τ(3)
sβClℐV=X for every nonempty set V∈sℐOX,τProof.
The proof follows from Theorem 14 of [16].Corollary 1.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is S∗-ℐ-hyperconnected(2)
V is ∗-dense for every nonempty set V∈sβℐOX,τ(3)
V is ∗-dense for every nonempty set V∈bℐOX,τ(4)
V is ∗-dense for every nonempty set V∈pℐOX,τ(5)
sClℐV=X for every nonempty set V∈pℐOX,τ(6)
pClℐV=X for every nonempty set V∈sℐOX,τ(7)
sβClℐV=X for every nonempty set V∈sℐOX,τProof.
The proof follows from Theorem 14 of [16].


## 4. Onθ-ℐ-Irreducible Ideal Topological Spaces

In this section, we introduce the notion ofθ-ℐ-irreducible ideal topological spaces. Moreover, some characterizations of θ-ℐ-irreducible ideal topological spaces are investigated.Definition 6.
A subsetA of an ideal topological space X,τ,ℐ is said to be(1)
R∗-ℐ-open if A=Int∗ClA(2)
R∗-ℐ-closed if its complement is R∗-ℐ-open
Byr∗ℐOX,τ (resp. r∗ℐCX,τ, we denote the family of all R∗-ℐ-open (resp. R∗-ℐ-closed) sets of an ideal topological space X,τ,ℐ.Definition 7.
An ideal topological spaceX,τ,ℐ is called θ-ℐ-irreducible if every pair of nonempty R∗-ℐ-closed sets of X has a nonempty intersection.Remark 2.
For an ideal topological spaceX,τ,ℐ, we have the following diagram:(7)X,τ,ℐis∗‐hyperconnected⟹X,τ,ℐisθ‐ℐ‐irreducible.
The implication in the diagram is not reversible as shown in the following example.Example 2.
LetX=a,b,c with a topology τ=∅,X,a,b,a,b and an ideal ℐ=∅. Then, the ideal topological space X,τ,ℐ is θ-ℐ-irreducible, but X,τ,ℐ is not ∗-hyperconnected.Definition 8 (see [25]).
A subsetA of an ideal topological space X,τ,ℐ is said to be θ-ℐ-closed if Clθ∗A=A, where(8)Clθ∗A=x∈XA∩Cl∗U≠∅for each open setUcontainingx.Theorem 10.
An ideal topological spaceX,τ,ℐ is θ-ℐ-irreducible if and only if Clθ∗Cl∗U=X for each nonempty open set U of X.Proof.
LetX,τ,ℐ be θ-ℐ-irreducible. Suppose that there exists a nonempty open set U such that Clθ∗Cl∗U≠X. Then, there exists a nonempty open set V such that Cl∗V∩Cl∗U=∅. Since Cl∗IntU and Cl∗IntV are R∗-ℐ-closed sets, X,τ,ℐ is not θ-ℐ-irreducible.
Conversely, letClθ∗Cl∗U=X for each nonempty open set U of X. Suppose that X,τ,ℐ is not θ-ℐ-irreducible. There exist nonempty R∗-ℐ-closed sets F1 and F2 such that F1∩F2=∅. Then, IntF1 and IntF2 are nonempty open sets. Since IntF1≠∅, there exists an open set G such that x∈G⊆F1, and hence Cl∗G⊆Cl∗F1. This implies that Cl⋆G∩Cl⋆F2⊆Cl∗F1∩Cl∗F2=Cl∗IntF1∩Cl∗IntF2=F1∩F2=∅. Thus, Cl∗G∩Cl∗F2=∅. Therefore, x∉Clθ∗Cl∗F2 and Clθ∗Cl∗IntF2=Clθ∗Cl∗F2≠X. This is a contradiction.Theorem 11.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is θ-ℐ-irreducible(2)
Cl∗U∩Cl∗V≠∅ for every nonempty U,V∈sβℐOX,τ(3)
Cl∗U∩Cl∗V≠∅ for every nonempty U,V∈sℐOX,τ(4)
Cl∗U∩Cl∗V≠∅ for every nonempty U,V∈τ(5)
Cl∗U∩Cl∗V≠∅ for every nonempty U,V∈s∗ℐOX,τProof.
1⟹2: let U and V be nonempty strong β-ℐ-open sets. Then, we have Cl∗U=Cl∗IntCl∗U and Cl∗V=Cl∗IntCl∗V. This implies that Cl∗U and Cl∗V are R∗-ℐ-closed sets. Since X,τ,ℐ is θ-ℐ-irreducible, Cl∗U∩Cl∗V≠∅.2⟹3 and 3⟹4 are obvious.4⟹5: for a nonempty set U∈s∗ℐOX,τ, Cl∗U=Cl∗IntCl∗U and IntU≠∅. By (4), we have ∅≠Cl∗IntU∩Cl∗IntV=Cl∗U∩Cl∗V for every nonempty U,V∈s∗ℐOX,τ.5⟹1: this is obvious since r∗ℐCX,τ⊆s∗ℐOX,τ.Definition 9.
LetA be a subset of an ideal topological space X,τ,ℐ. A point x∈X is called a θ-semi-ℐ-cluster point of A if A∩Cl∗U≠∅ for every semi-ℐ-open set U containing x. The set of all θ-semi-ℐ-cluster points of A is called the θ-semi-ℐ-closure of A and is denoted by sClθ∗A. A point x∈X is called a θ-semi-ℐ-interior point of A if there exists a semi-ℐ-open set U containing x such that Cl∗U⊆A. The set of all θ-semi-ℐ-interior points of A is called the θ-semi-ℐ-interior of A and is denoted by sIntθ∗A.Lemma 9.
LetX,τ,ℐ be an ideal topological space. Then, sClθ∗A⊆Clθ∗A for every subset A of X.Theorem 12.
For an ideal topological spaceX,τ,ℐ, the following properties are equivalent:(1)
X,τ,ℐ is θ-ℐ-irreducible(2)
sClθ∗Cl∗U=X for every nonempty U∈τ(3)
sClθ∗Cl∗U=X for every nonempty U∈sℐOX,τ(4)
Clθ∗Cl∗U=X for every nonempty U∈sℐOX,τProof.
1⟹2: suppose that there exists a nonempty open set U such that sClθ∗Cl∗U≠X. Then, there exists a nonempty semi-ℐ-open set V such that Cl∗U∩Cl∗V=∅. Since V∈sℐOX,τ, Cl∗V=Cl∗IntV∈r∗ℐCX,τ, and hence X,τ,ℐ is not θ-ℐ-irreducible.2⟹3: for a nonempty set U∈sℐOX,τ, we have IntU≠∅, and hence X=sClθ∗Cl∗IntU=sClθ∗Cl∗U.3⟹4: it follows from Lemma 9.4⟹1: since τ⊆sℐOX,τ, this is obvious by Theorem 10.Definition 10.
An ideal topological spaceX,τ,ℐ is said to be semi-ℐ-Urysohn if for each distinct points x and y of X, there exist U,V∈sℐOX,τ such that x∈U, y∈V and Cl∗U∩Cl∗V=∅.Definition 11.
A functionf:X,τ,ℐ⟶Y,σ,J is said to be θ-ℐ-irresolute if for each point x∈X and each semi-J-open set V containing fx, there exists a semi-ℐ-open set U containing x such that fCl∗U⊆Cl∗V.Theorem 13.
IfX,τ,ℐ is θ-ℐ-irreducible, Y,σ,J is semi-J-Urysohn, and f:X,τ,ℐ⟶Y,σ,J is θ-ℐ-irresolute, then f is constant.Proof.
Suppose that there exist distinct pointsx,y of X such that fx≠fy. Since Y,σ,J is semi-J-Urysohn, there exist U,V∈sJOY,σ such that fx∈U, fy∈V and Cl∗U∩Cl∗V=∅. By the θ-ℐ-irresoluteness of f, there exist G,W∈sℐOX,τ containing x and y, respectively, such that fCl∗G⊆Cl∗U and fCl∗W⊆Cl∗V. Thus, Cl∗G∩Cl∗W=∅. By Theorem 11, this contradicts the assumption that X,τ,ℐ is θ-ℐ-irreducible.Lemma 10.
Iff:X,τ,ℐ⟶Y,σ,J is θ-ℐ-irresolute, then(9)fsClθ∗A⊆sClθ∗fA,for every subset A of X.Proof.
LetA be a subset of X and x∉f−1sClθ∗fA. Then, fx∉sClθ∗fA, and there exists a semi-J-open set V containing fx such that Cl∗V∩fA=∅. Since f is θ-ℐ-irresolute, there exists a semi-ℐ-open set U containing x such that fCl∗U⊆Cl∗V. Thus, fCl∗U∩fA=∅ and Cl∗U∩A=∅. Therefore, x∉sClθ∗A. This shows that sClθ∗A⊆f−1sClθ∗fA. Consequently, we obtain fsClθ∗A⊆sClθ∗fA.Theorem 14.
Iff:X,τ,ℐ⟶Y,σ,J is a θ-ℐ-irresolute surjection and X,τ,ℐ is θ-ℐ-irreducible, then Y,σ,J is θ-J-irreducible.Proof.
Suppose thatX,τ,ℐ is θ-ℐ-irreducible. Let V be a nonempty semi-J-open set of Y. Since f is surjective, there exists a point x of X such that fx∈V. By the θ-ℐ-irresoluteness of f, there exists a semi-ℐ-open set U containing x such that fCl∗U⊆Cl∗V. Since X,τ,ℐ is θ-ℐ-irreducible, by Theorem 12, we have sClθ∗Cl∗U=X. By Lemma 10,(10)Y=fX=fsClθ∗Cl∗U⊆sClθ∗fCl∗U⊆sClθ∗Cl∗V.
It follows from Theorem12 that Y,σ,J is θ-J-irreducible.



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