Abstract
We study the product properties of nearly Lindelöf, almost Lindelöf, and weakly Lindelöf spaces. We prove that in weak -spaces, these topological properties are preserved under finite topological products. We also show that the product of separable spaces is weakly Lindelöf.
1. Introduction
In 1959 Frolík [1] introduced the notion of weakly Lindelöf space that afterward was studied by several authors. In 1982 Balasubramanian [2] introduced and studied the notion of nearly Lindelöf spaces as a generalization of the nearly compact spaces; then in 1986 Mršević et al. [3] gave some characterizations of these spaces. In 1984 Willard and Dissanayake [4] gave the notion of almost Lindelöf spaces. In 1996 Cammaroto and Santoro [5] studied and gave further new results related to these generalizations of Lindelöf spaces, and recently the authors (see [6–8]) studied mappings and semiregular property on these generalizations of Lindelöf spaces. By using the regularly open and regularly closed sets, these structures can also be extended to the bitopological spaces; for more details on regularly pairwise open and closed sets see, for example, [9–12].
It is well known that many of the results on the invariance of covering properties under product are negative, that is, the covering properties are simply not preserved by the product unless one or more of the factors are assumed to be satisfied as additional conditions.
In this work, we discuss the product problem in the sense of generalizations of Lindelöf spaces, namely, nearly Lindelöf, almost Lindelöf, and weakly Lindelöf spaces. We will note that a well-known example shows the properties nearly Lindelöf and almost Lindelöf are not finitely productive. We also give some necessary conditions for these covering properties to be preserved under a finite product.
In this paper, we let be a topological space on which no separation axioms are considered unless explicitly stated. The interior and the closure of any subset of will be denoted by and , respectively. Recall that a subset is called regularly open (regularly closed) if . The topology generated by regularly open subsets of a space is called the semiregularization of the space and is denoted by or simply by . A space is said to be semiregular if the regularly open sets form a base for the topology or equivalently . By regularly open cover of we mean a cover of by regularly open sets in . Moreover, a space is called a -space if every -set is open in , and it is called nearly compact [13] if every open cover of admits a finite subfamily such that or, equivalently, every regularly open cover of has a finite subcover.
Recall also that a function from a topological space to a topological space is called -map [14] (almost continuous [15]) if is regularly open (open) in for every regularly open set in . It is called -continuous [16] if for every and every open subset of containing , there exists an open subset in containing such that . Moreover, is called almost open [15] if is open in for every regularly open subset in , and it is called almost closed [15] if is closed in for every regularly closed subset in .
2. Preliminaries
It is known that a nonempty product space is Hausdorff (regular, completely regular, resp.) if and only if each factor space is Hausdorff (regular, completely regular, resp.). A nonempty product space is compact if and only if each factor space is compact. Moreover, the product of a paracompact space with a compact -space is paracompact. However, products of normal, paracompact, or Lindelöf spaces often fail to be normal, paracompact, or Lindelöf, respectively. Note that a space is a -space if and only if the countable union of closed sets is closed if and only if the countable intersection of open sets is open.
It is well known that the product of two Lindelöf spaces is not necessarily Lindelöf since the Sorgenfrey line is Lindelöf but the Sorgenfrey plane is not Lindelöf. In 1972, Misra [17] proved that, in -spaces, finite product of -spaces is a -space and no infinite product of -spaces with more than one point is a -space. We note that Misra's result for -spaces, that finite product of -spaces is a -space, also holds for arbitrary spaces. Misra [17] also proved that the product of two Lindelöf -spaces is a Lindelöf -space. Thus, in -spaces, finite product of Lindelöf spaces is Lindelöf. The following proposition shows that it is sufficient that one of the two spaces is a -space, to ensure that their product is Lindelöf. In fact, for -spaces, this result is an immediate corollary to Misra's Theorem 2.1 and Proposition 4.2(g) in [17].
Proposition 2.1. Let be a Lindelöf -space and a Lindelöf space. Then is Lindelöf.
Proof. Let be an open cover of ; Let ; then for each , there exists and such that . The subspace of is a homeomorphic copy of the Lindelöf space , thus it is Lindelöf. Now is an open cover of ; hence is an open cover of the Lindelöf space . Thus it has a countable subcollection which covers . So the countable family covers . Now define . Since is a -space, is an open subset in . Thus is a countable open cover of the slab , and hence all the more is a countable open cover of the slab .
Now the collection of sets is an open cover of the Lindelöf space . So it has a countable set of points such that . Thus .
Thus we have a countable collection of slabs covering , and for each , is covered by the countable collection . So is a countable subcover of . Thus is Lindelöf, which completes the proof.
Remark 2.2. (a) Recall that a space is a weak -space [18] if, for each countable family of open sets in , we have . Clearly, is a weak -space if and only if the countable union of regularly closed sets is regularly closed if and only if the countable intersection of regularly open sets is regularly open. Moreover, a space is a weak -space if and only if is a -space (see [19]). Note also that every -space is a weak -space but the converse is not necessarily true, since the finite complement topology on is a weak -space but it is not a -space.
(b) It is a known fact that the semiregularization of a product space is the product of the semiregularizations of the factor spaces.
Lemma 2.3. Finite product of weak -spaces is a weak -space.
Proof. We prove for only two spaces using Remark 2.2. So let and be two weak -spaces. Then and are -spaces. Thus is a -space. Therefore, is a weak -space.
Note that infinite product of weak -spaces is not necessarily a weak -space, since if is any discrete space containing more than one point and is infinite; then the product space is not a -space, and, since it is semiregular, it cannot be a weak -space either.
3. On Generalized Lindelöf Spaces
Definition 3.1 (see [1, 2, 4]). A topological space is called nearly Lindelöf, almost Lindelöf, and weakly Lindelöf if, for every open cover of , there exists a countable subset such that respectively.
One can easily show that if a space is semiregular and nearly Lindelöf (or regular and almost Lindelöf), then it is Lindelöf. And it is well known that the Sorgenfrey line is regular and Lindelöf, but the Sorgenfrey plane is not Lindelöf. Thus, neither of the properties almost Lindelöf and nearly Lindelöf is finitely productive.
The following proposition shows that if the product of topological spaces has any property of Definition 3.1, then each factor space has the same property.
Proposition 3.2. Suppose that is a nonempty topological space. If is nearly Lindelöf (resp., almost Lindelöf or weakly Lindelöf), then is nearly Lindelöf (resp., almost Lindelöf or weakly Lindelöf).
Proof. Since the projection map is a continuous and open function from onto , it is almost continuous and almost open. Thus is -continuous and -map (see [20, 21]). Therefore, is nearly Lindelöf, almost Lindelöf, and weakly Lindelöf (see [7, Corollary 3.1], [8, Corollary 3.3], and [22, Theorem 3.2], resp.).
In [5], it was shown that the product of a nearly Lindelöf space with a nearly compact space is nearly Lindelöf. Next we prove analogous results concerning almost Lindelöf and weakly Lindelöf spaces.
Proposition 3.3. Let be an almost Lindelöf (weakly Lindelöf) space and a nearly compact space. Then is almost Lindelöf (weakly Lindelöf).
Proof. The proof of Proposition 3.3 is similar to the proof of an analogous result for nearly Lindelöf spaces (see [5, Proposition 1.9]).
Note that a space is nearly Lindelöf if and only if is Lindelöf (see [3, Theorem 1]). Thus, using this fact, the proof of the following theorem becomes easy.
Theorem 3.4. The product of a nearly Lindelöf weak -space with a nearly Lindelöf space is nearly Lindelöf.
Proof. Let be a nearly Lindelöf weak -space and nearly Lindelöf. Thus, by Remark 2.2(a), is a Lindelöf -space and is Lindelöf. So, by Proposition 2.1, is Lindelöf. Therefore, is nearly Lindelöf.
Now on using Theorem 3.4 and Lemma 2.3, we conclude the following corollary.
Corollary 3.5. The product of finitely many nearly Lindelöf spaces, all but one of which are weak -spaces, is nearly Lindelöf.
Next we prove that the result in Theorem 3.4 above is correct for almost Lindelöf and weakly Lindelöf spaces.
Theorem 3.6. The product of an almost Lindelöf weak -space with an almost Lindelöf space is almost Lindelöf.
Proof. Since almost Lindelöf property is a semiregular property, that is, a space is almost Lindelöf if and only if is almost Lindelöf (see [6, Theorem 2.1]), it is sufficient to prove that is almost Lindelöf. Thus let be an open cover of , and, without loss of generality, suppose that for every where is regularly open in and is regularly open in . Fix , and, for each , there exists such that .
Now is an open cover of the almost Lindelöf space , so it has a countable subset such that . Put . Since is a weak -space, is a regularly open set in . Thus is an open cover of the almost Lindelöf space , so it has a countable subset such that . Therefore,
Since the last term is countable, thus is also almost Lindelöf and therefore is almost Lindelöf.
Corollary 3.7. The product of finitely many almost Lindelöf spaces, all but one of which are weak -spaces, is almost Lindelöf.
For weakly Lindelöf spaces we give the following results.
Theorem 3.8. The product of a weakly Lindelöf weak -space with a weakly Lindelöf space is weakly Lindelöf.
Proof. The proof of Theorem 3.8 is similar to the proof of Theorem 3.6, thus the details are omitted.
Corollary 3.9. The product of finitely many weakly Lindelöf spaces, all but one of which are weak -spaces, is weakly Lindelöf.
Next we prove that the product of separable spaces is weakly Lindelöf. First we recall that a space is called separable if it has a countable dense subset, and one says that has caliber if, whenever is a family of open subsets of with , a subfamily of exists with and . One also says that satisfies the countable chain condition if every family of disjoint open subsets of is countable. Moreover, is called almost rc-Lindelöf [23] if every regularly closed cover of has a countable subfamily whose union is dense in .
Theorem 3.10. The product of separable spaces is weakly Lindelöf.
Proof. Let be a family of separable spaces. Then has caliber . Thus satisfies the countable chain condition. So, is an almost rc-Lindelöf space (see [23, Proposition 2.2]). Therefore, is weakly Lindelöf (see [23, Theorem 2.1]).
Note that the product of two separable Lindelöf spaces need not be almost Lindelöf, since the Sorgenfrey line is separable, but is not almost Lindelöf since it is a regular non-Lindelöf space.
Since in weak -spaces, for any countable open subsets of , we have . Thus we conclude the following lemma.
Lemma 3.11. In weak -spaces, weakly Lindelöf property and almost Lindelöf property are equivalent.
So depending on Theorem 3.6 we conclude the following corollaries.
Corollary 3.12. If is a weakly Lindelöf weak -space and is almost Lindelöf, then is almost Lindelöf.
Corollary 3.13. In weak -spaces, finite product of weakly Lindelöf spaces is almost Lindelöf.
Acknowledgment
The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU and Fundamental Research Grant Scheme 01-11-09-723FR.