#### Abstract

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

#### 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 a -space if every nonempty open set of is dense in and showed that is a -space if and only if it is hyperconnected. Pipitone and Russo [3] defined a topological space to be semiconnected if is not the union of two disjoint nonempty semiopen sets of and showed that is semiconnected if and only if it is a -space. Sharma [4] indicated that a space is a -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]. -spaces, -connected spaces [8], semiconnected spaces, and irreducible spaces are equivalent to hyperconnected spaces. Janković and Long [9] have introduced and investigated the notion of -irreducible spaces. It is pointed out in [9] that hyperconnectedness implies -irreducibility and -irreducibility implies connectedness. Noiri and Umehara [10] investigated some characterizations and some preservation theorems concerning -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--open sets, --open sets, and --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--open sets and semi--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*--hyperconnected ideal topological spaces. In the last section, we introduce the notion of --irreducible ideal topological spaces. Especially, several characterizations of --irreducible ideal topological spaces are established.

#### 2. Preliminaries

Throughout the present paper, spaces and (or simply and ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. In a topological space , the closure and the interior of any subset of will be denoted by and , respectively. An ideal on a topological space is a nonempty collection of subsets of satisfying the following properties: (1) and imply ; (2) and imply . A topological space with an ideal on is called an ideal topological space and is denoted by . For an ideal topological space and a subset of , is defined as follows: for every open neighbourhood of . In case there is no chance for confusion, is simply written as . In [11], is called the local function of with respect to and and defines a Kuratowski closure operator for a topology . For every ideal topological space , there exists a topology finer than , generated by the base . However, is not always a topology [12]. A subset is said to be -closed [13] if . The interior of a subset in is denoted by .

*Definition 1. *A subset of an ideal topological space is said to be(1)Pre--open [17] if (2)Semi--open [14] if (3)--open [16] if (4)Strong --open [18] if (5)--open [19] if (6)--open [20] if (7)-Dense [21] if By (resp. , , , , and ), we denote the family of all pre--open (resp. semi--open, --open, strong --open, --open, and --open) sets of an ideal topological space .

The complement of a pre--open (resp. semi--open, --open, strong --open, and --open) set is called pre--closed [17] (resp. semi--closed [14], --closed [16, 22], strong --closed [18], and --closed [19]).

*Definition 2 (see [16]). *The pre--closure (resp. semi--closure, --closure, and strong --closure) of a subset of an ideal topological space , denoted by (resp. , , and ), is defined by the intersection of all pre--closed (resp. semi--closed, --closed, and strong --closed) sets of containing .

Lemma 1 (see [16]). *For a subset of an ideal topological space , the following properties hold:*(1)*(2)**(3)**(4)*

Lemma 2 (see [23]). *Let be a subset of an ideal topological space and be an open set. Then, .*

#### 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 space is said to be ∗-hyperconnected if is ∗-dense for every nonempty open set of .

The following theorems give several characterizations of ∗-hyperconnected ideal topological spaces.

Theorem 1. *For an ideal topological space , the following properties are equivalent:*(1)* is ∗-hyperconnected*(2)* for every nonempty set *(3)* for every nonempty set *

*Proof. * : suppose that is ∗-hyperconnected. Let be a nonempty pre--open set. Then, is strong --open; by Theorem 14 of [16], . : let be a nonempty pre--open set. By (2), , and hence . : let be a nonempty open set. Then, is pre--open, and by (3), . Thus, is -hyperconnected.

Lemma 3. *Let be an ideal topological space. Then, for every .*

*Proof. *It follows from Lemma 1(3) and Theorem 3.4(1) of [15].

Theorem 2. *For an ideal topological space , the following properties are equivalent:*(1)* is -hyperconnected*(2)* for every nonempty set and every nonempty set *(3)*The only subsets of which are both semi--open and --closed in are empty set and itself*(4)* cannot be expressed by the disjoint union of a nonempty semi--open set and a nonempty --open set*(5)* for every nonempty set *

*Proof. * : it follows from Theorem 11 of [16]. : let be a subset of , which is both semi--open and --closed. Then, is semi--open and is --open such that , which is a contradiction of (2). : suppose that , where is a nonempty semi--open set and a nonempty --open set such that . Since and is --open, is --closed. This shows that is both semi--open and --closed, which is a contradiction of (3). : suppose that for some nonempty set . Then, we have , and Since is --closed, is --open, and by Lemma 3, is semi--open, which is a contradiction of (4). : let be a nonempty open set. Then, is semi--open, and by (5), . Thus, . This shows that is -hyperconnected.Next, we will introduce and study the concept of --hyperconnected ideal topological spaces.

*Definition 4. *An ideal topological space is called *-**-hyperconnected* if for every nonempty open set .

*Remark 1. *For an ideal topological space , we have the following diagram:The implication in the diagram is not reversible as shown in the following example.

*Example 1 (see [16]). *Let with a topologyand an ideal . Then is -hyperconnected, but is not --hyperconnected.

Lemma 4 (see [15]). *A subset of an ideal topological space is semi--open if and only if there exists such that .*

Theorem 3. *For an ideal topological space , the following properties are equivalent:*(1)* is --hyperconnected*(2)* or for every subset of *(3)* for every nonempty open set and every nonempty --open set *(4)* for every nonempty semi--open set and every nonempty --open set *

*Proof. * : let be a subset of . Suppose that . Since is --hyperconnected, . Thus, . : suppose that for some nonempty open set and nonempty --open set . Then, , and by (2), . Moreover, since is open, . : suppose that for some nonempty semi--open set and nonempty --open set . Since is a nonempty semi--open set, by Lemma 4, there exists a nonempty open set such that . Then, is a nonempty semi--open set such that , which is a contradiction of (3). : suppose that is not --hyperconnected. Therefore, there exists a nonempty open set of such that , and hence . Thus, is a nonempty --open set and is a nonempty semi--open set such that . This is a contradiction. Consequently, we obtain is --hyperconnected.

Theorem 4. *For an ideal topological space , the following properties are equivalent:*(1)* is -hyperconnected*(2)* for every nonempty set *(3)* for every nonempty set *(4)* is --hyperconnected*

*Proof. * and follow from Theorem 1. and are obvious.For a subset of an ideal topological space , we denote by the relative topology on and is an ideal on [24].

Lemma 5 (see [24]). *Let be an ideal topological space and . Then, .*

Lemma 6 (see [23]). *Let be an ideal topological space and . Then, .*

Lemma 7. *Let be an open set of an ideal topological space . If , then .*

*Proof. *Suppose that is an open set and . Since and is open in , there exists an open set such that . Since and by Lemma 2,Thus, .

Lemma 8. *Let be an open set of an ideal topological space . Then, for every .*

*Proof. *Let be a subset of . Then, we have

Theorem 5. *Let be an open set of an ideal topological space . If is --hyperconnected, then is --hyperconnected.*

*Proof. *Suppose that is --hyperconnected. Let be a nonempty pre--open set of . By Lemma 7, is pre--open in . Since is --hyperconnected, , and by Lemma 8, . Therefore, is --hyperconnected.

Theorem 6. *For an ideal topological space , the following properties are equivalent:*(1)* is --hyperconnected*(2)* for every nonempty open set *

*Proof. * : it follows from Theorem 14 of [16]. : let be a nonempty open set. By (2), we haveand hence . This shows that is --hyperconnected.

*Definition 5. *A function is said to be --continuous if for each point and each open set containing , there exists an open set containing such that .

Theorem 7. *If is a --continuous surjection and is --hyperconnected, then is --hyperconnected.*

*Proof. *Suppose that is --hyperconnected. Let be a nonempty open set of . Since is surjective, there exists a point of such that . Since is --continuous, there exists an open set containing such that . Since is --hyperconnected, by Theorem 6, we have , and hence . It follows from Theorem 6 that is --hyperconnected.

Theorem 8. *For an ideal topological space , the following properties are equivalent:*(1)* is --hyperconnected*(2)* is -dense for every nonempty set *(3)* for every nonempty set *

*Proof. *The proof follows from Theorem 14 of [16].

Theorem 9. *For an ideal topological space , the following properties are equivalent:*(1)* is --hyperconnected*(2)* is -dense for every nonempty set *(3)* for every nonempty set *

*Proof. *The proof follows from Theorem 14 of [16].

Corollary 1. *For an ideal topological space , the following properties are equivalent:*(1)* is --hyperconnected*(2)* is -dense for every nonempty set *(3)* is -dense for every nonempty set *(4)* is -dense for every nonempty set *(5)* for every nonempty set *(6)* for every nonempty set *(7)* for every nonempty set *

*Proof. *The proof follows from Theorem 14 of [16].

#### 4. On --Irreducible Ideal Topological Spaces

In this section, we introduce the notion of --irreducible ideal topological spaces. Moreover, some characterizations of --irreducible ideal topological spaces are investigated.

*Definition 6. *A subset of an ideal topological space is said to be(1)--open if (2)--closed if its complement is --openBy (resp. , we denote the family of all --open (resp. --closed) sets of an ideal topological space .

*Definition 7. *An ideal topological space is called --*irreducible* if every pair of nonempty --closed sets of has a nonempty intersection.

*Remark 2. *For an ideal topological space , we have the following diagram:The implication in the diagram is not reversible as shown in the following example.

*Example 2. *Let with a topology and an ideal . Then, the ideal topological space is --irreducible, but is not -hyperconnected.

*Definition 8 (see [25]). *A subset *A* of an ideal topological space is said to be --closed if , where

Theorem 10. *An ideal topological space is --irreducible if and only if for each nonempty open set of .*

*Proof. *Let be --irreducible. Suppose that there exists a nonempty open set such that . Then, there exists a nonempty open set such that . Since and are --closed sets, is not --irreducible.

Conversely, let for each nonempty open set of . Suppose that is not --irreducible. There exist nonempty --closed sets and such that . Then, and are nonempty open sets. Since , there exists an open set such that , and hence . This implies that . Thus, . Therefore, and . This is a contradiction.

Theorem 11. *For an ideal topological space , the following properties are equivalent:*(1)* is --irreducible*(2)* for every nonempty *(3)* for every nonempty *(4)* for every nonempty *(5)* for every nonempty *

*Proof. * : let and be nonempty strong --open sets. Then, we have and . This implies that and are --closed sets. Since is --irreducible, . and are obvious. : for a nonempty set , and . By (4), we have for every nonempty . : this is obvious since .

*Definition 9. *Let be a subset of an ideal topological space . A point is called a -*semi*--*cluster point* of if for every semi--open set containing . The set of all -semi--cluster points of is called the -*semi*--*closure* of and is denoted by . A point is called a -*semi*--*interior point* of if there exists a semi--open set containing such that . The set of all -semi--interior points of is called the -*semi*--*interior* of and is denoted by .

Lemma 9. *Let be an ideal topological space. Then, for every subset of .*

Theorem 12. *For an ideal topological space , the following properties are equivalent:*(1)* is --irreducible*(2)* for every nonempty *(3)* for every nonempty *(4)* for every nonempty *

*Proof. * : suppose that there exists a nonempty open set such that . Then, there exists a nonempty semi--open set such that . Since , , and hence is not --irreducible. : for a nonempty set , we have , and hence . : it follows from Lemma 9. : since , this is obvious by Theorem 10.

*Definition 10. *An ideal topological space is said to be *semi*--*Urysohn* if for each distinct points and of , there exist such that , and .

*Definition 11. *A function is said to be --*irresolute* if for each point and each semi--open set containing , there exists a semi--open set containing such that .

Theorem 13. *If is --irreducible, is semi--Urysohn, and is --irresolute, then is constant.*

*Proof. *Suppose that there exist distinct points of such that . Since is semi--Urysohn, there exist such that , and . By the --irresoluteness of , there exist containing and , respectively, such that and . Thus, . By Theorem 11, this contradicts the assumption that is --irreducible.

Lemma 10. *If is --irresolute, thenfor every subset of .*

*Proof. *Let be a subset of and . Then, , and there exists a semi--open set containing such that . Since is --irresolute, there exists a semi--open set containing such that . Thus, and . Therefore, . This shows that . Consequently, we obtain .

Theorem 14. *If is a --irresolute surjection and is --irreducible, then is --irreducible.*

*Proof. *Suppose that is --irreducible. Let be a nonempty semi--open set of . Since is surjective, there exists a point of such that . By the --irresoluteness of , there exists a semi--open set containing such that . Since is --irreducible, by Theorem 12, we have . By Lemma 10,It follows from Theorem 12 that is --irreducible.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.

#### Acknowledgments

This research project was financially supported by Mahasarakham University.