Journal of Mathematics
Volume 2020 (2020), Article ID 9387601, 7 pages
https://doi.org/10.1155/2020/9387601

Research Article

On Characterizations of ∗-Hyperconnected Ideal Topological Spaces

Chawalit Boonpok

Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand

Correspondence should be addressed to Chawalit Boonpok; chawalit.b@msu.ac.th

Received 20 April 2020; Accepted 17 August 2020; Published 22 September 2020

Academic Editor: Ali Jaballah

Copyright © 2020 Chawalit Boonpok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 topology

and 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 have

and 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 --open

By (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, then

for 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.

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