Adem Kılıçman¹ and Zabidin Salleh²

Abstract

We will introduce and study the pairwise weakly regular-Lindelöf bitopological spaces and obtain some results. Furthermore, we study the pairwise weakly regular-Lindelöf subspaces and subsets, and investigate some of their characterizations. We also show that a pairwise weakly regular-Lindelöf property is not a hereditary property. Some counterexamples will be considered in order to establish some of their relations.

1. Introduction

The study of bitopological spaces was first initiated by Kelly [1] in 1963 and thereafter a large number of papers have been done to generalize the topological concepts to bitopological setting. In literature, there are several generalizations of the notion of Lindelöf spaces, and these are studied separately for different reasons and purposes. In 1959, Frolík [2] introduced the notion of weakly Lindelöf spaces and in 1996, Cammaroto and Santoro [3] studied and gave further new results about these spaces followed by Klçman and Fawakhreh [4]. In the same paper, Cammaroto and Santoro introduced the notion of weakly regular-Lindelöf spaces by using regular covers and leave open the study of this new concept. In 2001, Fawakhreh and Klçman [5] studied this new generalization of Lindelöf spaces and obtained some results. Then, Klçman and Fawakhreh [6] studied subspaces of this spaces and obtained some results.

Recently, the authors studied pairwise Lindelöfness in [7] and introduced and studied the notion of pairwise weakly Lindelöf spaces in bitopological spaces, see [8], where the authors extended some results that were due to Cammaroto and Santoro [3], Klçman and Fawakhreh [4], and Fawakhreh [9]. In [10], the authors also studied the mappings and pairwise continuity on pairwise Lindelöf bitopological spaces. The purpose of this paper is to define the notion of weakly regular-Lindelöf property in bitopological spaces, which we will call pairwise weakly regular- spaces and investigate some of their characterizations. Moreover, we study the pairwise weakly regular-Lindelöf subspaces and subsets and also investigate some of their characterizations.

In Section 3, we will introduce the concept of pairwise weakly regular-Lindelöf bitopological spaces by using pairwise regular cover. This study begin by investigating the -weakly regular-Lindelöf property and some results obtained. Furthermore, we study the relation between -nearly Lindelöf, -almost Lindelöf, -weakly Lindelöf, -almost regular-Lindelöf, -nearly regular-Lindelöf, and -weakly regular-Lindelöf spaces, where or , .

In Section 4, we will define the concept of pairwise weakly regular-Lindelöf subspaces and subsets. We will define the concept of pairwise weakly regular-Lindelöf relative to a bitopological space by investigating the -weakly regular-Lindelöf property and obtain some results. The main result obtained is pairwise, and weakly regular-Lindelöf property is not a hereditary property by a counterexample given.

2. Preliminaries

Throughout this paper, all spaces and (or simply ) are always mean topological spaces and bitopological spaces, respectively, unless explicitly stated. We always use - to denote the certain properties with respect to topology and , where and . By - and -, we will mean the interior and the closure of a subset of with respect to topology , respectively. We denote by and for the interior and the closure of a subset of with respect to topology for each , respectively.

If , then - and - will be used to denote the interior and closure of with respect to topology in the subspace , respectively. By -open cover of , we mean that the cover of by -open sets in ; similar for the -regular open cover of and so forth. We will use the notation, is -Lindelöf space which mean that is a Lindelöf space, where .

Definition 2.1 (See [11, 12]). A subset of a bitopological space is said to be -regular open (resp., -regular closed) if - (resp., -, and is said pairwise regular open (resp., pairwise regular closed) if it is both -regular open and -regular open (resp., -regular closed and -regular closed).

Definition 2.2. Let be a bitopological space. A subset of is said to be

(i)-open if is open with respect to in , is said open in if it is both -open and -open in , or equivalently, ;(ii)-closed if is closed with respect in , is said closed in if it is both -closed and -closed in , or equivalently, ;(iii)-clopen if is both -closed and -open set in , is said clopen in if it is both -clopen and -clopen in (iv)-clopen if is -closed and -open set in , is said clopen if it is both -clopen and -clopen in .

Definition 2.3 (See [13]). A bitopological space is said to be Lindelöf if the topological space and are both Lindelöf. Equivalently, is Lindelöf if every -open cover of has a countable subcover for each .

Definition 2.4 (See [1, 11]). A bitopological space is said to be -regular if for each point and for each -open set of containing there exists an -open set such that - and is said to be pairwise regular if it is both -regular and -regular.

Definition 2.5 (See [11, 14]). A bitopological space is said to be -almost regular if for each and for each -regular open set of containing there is an -regular open set such that - then is said to be pairwise almost regular if it is both -almost regular and -almost regular.

Definition 2.6 (See [11, 12]). A bitopological space is said to be -semiregular if for each and for each -open set of containing there is an -open set such that - and is said pairwise semiregular if it is both -semiregular and -semiregular.

Definition 2.7. A bitopological space is said to be -nearly Lindelöf [15] (resp., -almost Lindelöf [16], -weakly Lindelöf [8]) if for every -open cover of there exists a countable subset of such that (2.1)and is said pairwise nearly Lindelöf (resp., pairwise almost Lindelöf, pairwise weakly Lindelöf) if it is both -nearly Lindelöf (resp., -almost Lindelöf, -weakly Lindelöf) and -nearly Lindelöf (resp., -almost Lindelöf, -weakly Lindelöf).

Definition 2.8 (See [8]). A subset of a bitopological space is said to be -weakly Lindelöf relative to if for every cover of by -open subsets of such that there exists a countable subset of such that -. is said pairwise weakly Lindelöf relative to if it is both -weakly Lindelöf relative to and -weakly Lindelöf relative to .

Definition 2.9 (See [8]). A bitopological space is said to be -nearly paracompact if every cover of by -regular open sets admits a locally finite refinement. is said pairwise nearly paracompact if it is both -nearly paracompact and -nearly paracompact.

3. Pairwise Weakly Regular-Lindelöf Spaces

Definition 3.1 (See [17]). An -open cover of a bitopological space is said to be -regular cover if for every there exists a nonempty -regular closed subset of such that and -. is said pairwise regular cover if it is both -regular cover and -regular cover.

Definition 3.2. A bitopological space is said to be -almost regular-Lindelöf [17] (resp., -nearly regular-Lindelöf [18]) if for every -regular cover of there exists a countable subset of such that (3.1)then is said pairwise almost regular-Lindelöf (resp., pairwise nearly regular-Lindelöf) if it is both -almost regular-Lindelöf (resp., -nearly regular-Lindelöf) and -almost regular-Lindelöf (resp., -nearly regular-Lindelöf).

Definition 3.3. A bitopological space is said to be -weakly regular-Lindelöf if for every -regular cover of , there exists a countable subset of such that -. is said pairwise weakly regular-Lindelöf if it is both -weakly regular-Lindelöf and -weakly regular-Lindelöf.

Obviously, every -weakly Lindelöf space is -weakly regular-Lindelöf, and every -almost regular-Lindelöf space is -weakly regular-Lindelöf.

The authors expected that the answer of these questions is no. We can answer Question 1. by some restrictions on the space with the following proposition. First of all, we need the following lemmas.

Lemma 3.4 (See [17]). Let be an -almost regular space. Then, for each and for each -regular open subset of containing there exist two -regular open subsets and of such that - - .

Lemma 3.5 (See [17]). A space is -regular if and only if it is -almost regular and -semiregular.

Proposition 3.6. An -weakly regular-Lindelöf and -regular space is -weakly Lindelöf.

Proof. Let be an -regular open cover of . For each , there exists such that . Since is -almost regular, there exist two -regular open subsets and of such that -- by Lemma 3.4. Since for each , there exists a -regular closed set - in such that - and -, the family is an -regular cover of . Since is -weakly regular-Lindelöf, there exists a countable set of points of such that --. So, - and since is -semiregular, therefore is -weakly Lindelöf.

Corollary 3.7. A pairwise weakly regular-Lindelöf and pairwise regular space is pairwise weakly Lindelöf.

Proposition 3.6 implies the following corollaries.

Corollary 3.8. Let be an -regular space. Then, is -weakly regular-Lindelöf if and only if it is -weakly Lindelöf.

Corollary 3.9. Let be a pairwise regular space. Then, is pairwise weakly regular-Lindelöf if and only if it is pairwise weakly Lindelöf.

Definition 3.10 (See [8]). A bitopological space is called -weak -space if for each countable family of -open sets in , we have -- then is called pairwise weak -space if it is both -weak -space and -weak -space.

The following proposition shows that in -weak -spaces, -almost regular-Lindelöf property equivalent to -weakly regular-Lindelöf property.

Proposition 3.11. Let be an -weak -spaces. Then, is -almost regular-Lindelöf if and only if is -weakly regular-Lindelöf.

Proof. The proof follows immediately from the fact that in -weak -spaces, -- for any countable family of -open sets in .

Corollary 3.12. Let be a pairwise weak -spaces. Then, is pairwise almost regular-Lindelöf if and only if is pairwise weakly regular-Lindelöf.

If is an -almost regular space, then is -almost regular-Lindelöf if and only if it is -nearly Lindelöf (see [17]). Thus, we have the following corollary.

Corollary 3.13. In -almost regular and -weak -spaces, -weakly regular-Lindelöf property is equivalent to -nearly Lindelöf property.

Proof. This is a direct consequence of Proposition 3.11 and the previous fact.

Corollary 3.14. In pairwise almost regular and pairwise weak -spaces, pairwise weakly regular-Lindelöf property is equivalent to pairwise nearly Lindelöf property.

Lemma 3.15 (See [17]). An -regular and -almost regular-Lindelöf space is -Lindelöf.

Corollary 3.16. In -regular and -weak -spaces, -weakly regular-Lindelöf property is equivalent to -Lindelöf property.

Proof. This is a direct consequence of Proposition 3.11 and Lemma 3.15.

Corollary 3.17. In pairwise regular and pairwise weak -spaces, pairwise weakly regular-Lindelöf property is equivalent to Lindelöf property.

Definition 3.18 (See [8]). A subset of a bitopological space is said to be -dense in or is an -dense subset of if -. is said dense in or is a dense subset of if it is -dense in or is an -dense subset of for each .

Definition 3.19 (See [8]). A bitopological space is said to be -separable if there exists a countable -dense subset of . is said separable if it is -separable for each .

Lemma 3.20 (See [8]). If the bitopological space is -separable, then it is -weakly Lindelöf.

Lemma 3.21 (See [18]). An -regular and -nearly regular-Lindelöf space is -Lindelöf.

It is clear that every -nearly regular-Lindelöf is -weakly regular-Lindelöf and every -almost regular-Lindelöf space is -weakly regular-Lindelöf, but the converses are not true in general as the following example show.

Example 3.22. Let be the collection of closed-open intervals in the real line : (3.2) Hence, is a base for the lower limit topology on . Choose usual topology as topology on . Thus, is a Lindelöf bitopological space (see [19]). Note that, sets of the form or are both -open and -closed in , and sets of the form and are -open in (see [19]). It is easy to check that is -regular since for each and for each -open set of the form in containing , there exists a -open set with such that -. We left to the reader to check for other forms of -open sets in . It is clear that is -separable since the rational numbers are a countable -dense subset of . So is -regular and -separable. Thus, is -weakly Lindelöf by Lemma 3.20, and so is -weakly regular-Lindelöf. It is known that is not -Lindelöf since the -closed subspace is not -Lindelöf for it is a discrete subspace (see [19]). Since is -regular, but not -Lindelöf, then it is neither -almost regular-Lindelöf nor -nearly regular-Lindelöf by Lemmas 3.15 and 3.21.

It is clear that every -almost Lindelöf is -weakly Lindelöf, but the converse is not true as in the following example show.

Lemma 3.23 (See [16]). An -regular space is -almost Lindelöf if and only if it is -Lindelöf.

Example 3.24. Let be a bitopological space defined as in Example 3.22 above. Example 3.22 shows that is -weakly Lindelöf, but not -Lindelöf. Since is -regular, but not -Lindelöf, then it is nor -almost Lindelöf by Lemma 3.23.

Remark 3.25. Example 3.24 solves the open problem in [8, Question 1].

Lemma 3.26 (See [8]). An -weakly Lindelöf, -regular, and -nearly paracompact bitopological space is -Lindelöf.

Proposition 3.27. Let be an -regular and -nearly paracompact spaces. Then, is -Lindelöf if and only if is -weakly regular-Lindelöf.

Proof. Let be an -regular, -nearly paracompact, and -weakly regular-Lindelöf space. Then, is -weakly Lindelöf by Proposition 3.6. So is -Lindelöf by Lemma 3.26. The converse is obvious.

Corollary 3.28. Let be a pairwise regular and pairwise nearly paracompact spaces. Then, is Lindelöf if and only if is pairwise weakly regular-Lindelöf.

Now, we give a characterization of -weakly regular-Lindelöf spaces.

Theorem 3.29. A bitopological spaces is -weakly regular-Lindelöf if and only if for every family of -closed subsets of such that for each there exists a -open subset of with and - , there exists a countable subfamily such that - .

Proof. Let be a family of -closed subsets of such that for each there exists a -open subset of with and . It follows that -. Since --, then --, that is, --. Therefore,(3.3)So and the family is an -regular cover of . Since is -weakly regular-Lindelöf, there exists a countable subfamily such that(3.4)Therefore, -.

Conversely, let be an -regular cover of . By Definition 3.1, for each , is -open set in and there exists a -regular closed subset of such that and -. The family of -closed subsets of is satisfying the condition, for each there exists a -open subset of such that and (3.5)By hypothesis, there exists a countable subset of such that -, that is, -. So - and, therefore, -. This completes the proof.

Corollary 3.30. A bitopological spaces is pairwise weakly regular-Lindelöf if and only if for every family of closed subsets of such that for each there exists an open subset of with and , there exists a countable subfamily such that .

The following diagram illustrates the relationship among the generalizations of pairwise Lindelöf spaces and the generalizations of pairwise regular-Lindelöf spaces in terms of -: (3.6)

4. Pairwise Weakly Regular-Lindelöf Subspaces and Subsets

A subset of a bitopological space is said to be -weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) if is -weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) as a subspace of , that is, is -weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) with respect to the inducted bitopology from the bitopology of .

Definition 4.1 (See [17]). Let be a subset of a bitopological space . A cover of by -open subsets of such that is said to be -regular cover of by -open subsets of if for each , there exists a nonempty -regular closed subset of such that and -. is said pairwise regular cover by open subsets of if it is both -regular cover of by -open subsets of and -regular cover of by -open subsets of .

Definition 4.2 (See [17]). A subset of a bitopological space is said to be -almost regular-Lindelöf relative to if for every -regular cover of by -open subsets of there exists a countable subset of such that -. is said pairwise almost regular-Lindelöf relative to if it is both -almost regular-Lindelöf relative to and -almost regular-Lindelöf relative to .

Definition 4.3. A subset of a bitopological space is said to be -weakly regular-Lindelöf relative to if for every -regular cover of by -open subsets of there exists a countable subset of such that -. is said pairwise weakly regular-Lindelöf relative to if it is both -weakly regular-Lindelöf relative to and -weakly regular-Lindelöf relative to .

Obviously, every -weakly Lindelöf relative to the space is -weakly regular-Lindelöf relative to the space and every -almost regular-Lindelöf relative to the space is -weakly regular-Lindelöf relative to the space.

The authors expected that the answer of both questions is no.

Theorem 4.4. A subset of a bitopological spaces is -weakly regular-Lindelöf relative to if and only if for every family of -closed subsets of such that for each there exists a -open subset of with and there exists a countable subfamily such that .

Proof. Let be a family of -closed subsets of such that for each there exists a -open subset of with and . It follows that -. Since --, then --, that is, --. Therefore, -. So - is a -regular closed subset of satisfying the condition of Definition 4.1. Thus, the family is an -regular cover of by -open subsets of . Since is -weakly regular-Lindelöf relative to , there exists a countable subfamily such that(4.1)Therefore, .

Conversely, let be an -regular cover of by -open subsets of . By Definition 4.1, for each there exists a -regular closed subset of such that and -. The family of -closed subsets of is satisfying the condition, for each there exists a -open set with(4.2)then it follows that, . By hypothesis, there exists a countable subset of such that(4.3)Thus we have, and, therefore, -. This completes the proof.

Corollary 4.5. A subset of a bitopological spaces is pairwise weakly regular-Lindelöf relative to if and only if for every family of closed subsets of such that for each there exists an open subset of with and , there exists a countable subfamily such that .

Proposition 4.6. A subset of a space is -weakly regular-Lindelöf relative to if and only if for every family of -regular open subsets of satisfying the conditions and for each there exists a nonempty -regular closed subset of such that and - , then there exists a countable subset of such that - .

Proof. The necessity is obvious by the Definitions 4.1 and 4.2 since every -regular open set in is -open. For the sufficiency, let be a family of -open sets in satisfying the conditions of Definition 4.1 above. Then is a family of -regular open sets in satisfying the conditions of the theorem, since for each we have -. By hypothesis, there exists a countable subset of such that(4.4)This implies that is -weakly regular-Lindelöf relative to and completes the proof.

Corollary 4.7. A subset of a space is pairwise weakly regular-Lindelöf relative to if and only if for every family of pairwise regular open subsets of satisfying the conditions and for each there exists a nonempty pairwise regular closed subset of such that and , then there exists a countable subset of such that .

Proposition 4.8. If is a countable family of subsets of a space such that each is -weakly regular-Lindelöf relative to , then is -weakly regular-Lindelöf relative to .

Proof. Let be an -regular cover of by -open subsets of . Then for each , there exists a nonempty -regular closed subset of such that and -. Let , then for each there exists a nonempty -regular closed subset of such that and -. So is an -regular cover of by -open subsets of . Since is -weakly regular-Lindelöf relative to , there exists a countable subfamily such that -. But a countable union of countable sets is countable, so(4.5)This implies that is -weakly regular-Lindelöf relative to and completes the proof.

Corollary 4.9. If is a countable family of subsets of a space such that each is pairwise weakly regular-Lindelöf relative to , then is pairwise weakly regular-Lindelöf relative to .

Proposition 4.10. If is an -weakly regular-Lindelöf subspace of a bitopological space , then is -weakly regular-Lindelöf relative to .

Proof. Let be an -regular cover of by -open subsets of . Then, for each there exists a nonempty -regular closed subset of such that and -. For each , we have - and are -open sets in , and is -closed set in . Since for each , there exists a -regular closed set - in such that - and(4.6)that is, -, then the family is an -regular cover of . Since is an -weakly regular-Lindelöf subspace of , there exists a countable subset of such that(4.7)This shows that is -weakly regular-Lindelöf relative to .

Corollary 4.11. If is a pairwise weakly regular-Lindelöf subspace of a bitopological space , then is pairwise weakly regular-Lindelöf relative to .

The authors expected that the answer is no.

Theorem 4.12. If every -regular closed proper subset of a bitopological space is -weakly regular-Lindelöf relative to , then is -weakly regular-Lindelöf.

Proof. Let be an -regular cover of . For each , there exists a nonempty -regular closed subset of such that and -. Fix an arbitrary and let . Put , then is an -regular closed subset of and -. Therefore, is an -regular cover of by -open subsets of by Definition 4.1. By hypothesis, is -weakly regular-Lindelöf relative to , hence there exists a countable subset of such that -. So, we have(4.8)So - and this shows that is -weakly regular-Lindelöf.

Corollary 4.13. If every pairwise regular closed proper subset of a bitopological space is pairwise weakly regular-Lindelöf relative to , then is pairwise weakly regular-Lindelöf.

It is very clear that Theorem 4.12 implies the following corollaries.

Corollary 4.14. If every -regular closed subset of a bitopological space is -weakly regular-Lindelöf relative to , then is -weakly regular-Lindelöf.

Corollary 4.15. If every pairwise regular closed subset of a bitopological space is pairwise weakly regular-Lindelöf relative to , then is pairwise weakly regular-Lindelöf.

Note that, the space in above propositions is any bitopological space. If we consider itself is an -weakly regular-Lindelöf, we have the following results.

Theorem 4.16. Let be an -weakly regular-Lindelöf space. If is a proper -clopen subset of , then is -weakly regular-Lindelöf relative to .

Proof. Let be an -regular cover of by -open subsets of . Hence the family is an -regular cover of since is a proper -clopen subset of is also a -regular closed subset of . Since is -weakly regular-Lindelöf, there exists a countable subfamily such that(4.9)But and are disjoint; therefore, we have -. This completes the proof.

Corollary 4.17. Let be a pairwise weakly regular-Lindelöf space. If is a proper clopen subset of , then is pairwise weakly regular-Lindelöf relative to .

It is very clear that Theorem 4.16 implies the following corollary.

Corollary 4.18. Let be an -weakly regular-Lindelöf space. If is an -clopen subset of , then is -weakly regular-Lindelöf relative to .

Corollary 4.19. Let be a pairwise weakly regular-Lindelöf space. If is a clopen subset of , then is pairwise weakly regular-Lindelöf relative to .

Question 6. Is -closed subspace of an -weakly regular-Lindelöf space -weakly regular-Lindelöf?

Question 7. Is -regular closed subspace of an -weakly regular-Lindelöf space -weakly regular-Lindelöf?

The authors expected that the answer of both questions is no. Observe that the condition in Theorem 4.16 that a subset should be -clopen is necessary and it is not sufficient to be only -open or -regular open as example below shows. Arbitrary subspaces of -weakly regular-Lindelöf spaces need not be -weakly regular-Lindelöf nor -weakly regular-Lindelöf relative to the spaces. An -open or -regular open subset of an -weakly regular-Lindelöf space is neither -weakly regular-Lindelöf nor -weakly regular-Lindelöf relative to the spaces as in the following example also show. We need the following lemma (see [20, page 11]).

Lemma 4.20. If is a countable subset of ordinals not containing , where being the first uncountable ordinal, then .

Example 4.21. Let denote the set of ordinals which are less than or equal to the first uncountable ordinal number , that is, . This is an uncountable well-ordered set with a largest element , having the property that if with , then is countable. Since is a totally ordered space, it can be provided with its order topology. Let us denote this order topology by . Choose discrete topology as another topology for denoted by . So form a bitopological space. Now it is known that is a -Lindelöf space [20], so it is -weakly Lindelöf and thus -weakly regular-Lindelöf. The subspace , however, is not -Lindelöf (see [20]). We notice that is -open subspace of and also -regular open subset of . Observe that is not -weakly regular-Lindelöf by Corollary 3.16 since it is -regular and -weak -space. Moreover, is not -weakly regular-Lindelöf relative to . In fact, the family of -open sets in is -regular cover of by -open subsets of because and for each , there exists a nonempty -regular closed subset of such that and -. But the family has no countable subfamily such that -. For if satisfy the condition: -closures of unions of it elements cover , then which is impossible by Lemma 4.20.

So we can conclude that an -weakly regular-Lindelöf property is not hereditary property and, therefore, pairwise weakly regular-Lindelöf property is not so.

Acknowledgments

The authors gratefully acknowledge the Ministry of Higher Education, Malaysia, and University Putra Malaysia (UPM) that this research was partially supported under the Fundamental Grant Project 01-01-07-158FR.

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