Product Property on Generalized Lindelöf Spaces

We study the product properties of nearly Lindelof, almost Lindelof, and weakly Lindelof spaces. We prove that in weak 𝑃-spaces, these topological properties are preserved under finite topological products. We also show that the product of separable spaces is weakly Lindelof.


Introduction
In 1959 Frolík 1 introduced the notion of weakly Lindel öf space that afterward was studied by several authors.In 1982 Balasubramanian 2 introduced and studied the notion of nearly Lindel öf 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 Lindel öf spaces.In 1996 Cammaroto and Santoro 5 studied and gave further new results related to these generalizations of Lindel öf spaces, and recently the authors see 6-8 studied mappings and semiregular property on these generalizations of Lindel öf 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 Lindel öf spaces, namely, nearly Lindel öf, almost Lindel öf, and weakly Lindel öf spaces.We will note that a well-known example shows the properties nearly Lindel öf and almost Lindel öf are not finitely productive.We also give some necessary conditions for these covering properties to be preserved under a finite product.

ISRN Mathematical Analysis
In this paper, we let X, τ be a topological space on which no separation axioms are considered unless explicitly stated.The interior and the closure of any subset A of X, τ will be denoted by Int A and Cl A , respectively.Recall that a subset

Preliminaries
It is known that a nonempty product space α∈Δ X α is Hausdorff regular, completely regular, resp.if and only if each factor space X α is Hausdorff regular, completely regular, resp. .A nonempty product space α∈Δ X α is compact if and only if each factor space X α is compact.Moreover, the product of a paracompact space with a compact T 2 -space is paracompact.However, products of normal, paracompact, or Lindel öf spaces often fail to be normal, paracompact, or Lindel öf, respectively.Note that a space X is a P -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 Lindel öf spaces is not necessarily Lindel öf since the Sorgenfrey line E is Lindel öf but the Sorgenfrey plane E × E is not Lindel öf.In 1972, Misra 17 proved that, in T 1 -spaces, finite product of P -spaces is a P -space and no infinite product of P -spaces with more than one point is a P -space.We note that Misra's result for T 1 -spaces, that finite product of P -spaces is a P -space, also holds for arbitrary spaces.Misra 17 also proved that the product of two Lindel öf P -spaces is a Lindel öf P -space.Thus, in P -spaces, finite product of Lindel öf spaces is Lindel öf.The following proposition shows that it is sufficient that one of the two spaces is a P -space, to ensure that their product is Lindel öf.In fact, for T 1 -spaces, this result is an immediate corollary to Misra's Theorem 2.1 and Proposition 4.2 g in 17 .
Proof.Let U {T α ×U α : α ∈ Δ} be an open cover of X×Y ; Let x ∈ X; then for each y ∈ Y , there exists T α x y ∈ τ and U α x y ∈ σ such that x, y ∈ T α x y × U α x y .The subspace   19 .Note also that every Pspace is a weak P -space but the converse is not necessarily true, since the finite complement topology on R is a weak P -space but it is not a P -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 P -spaces is a weak P -space.
Proof.We prove for only two spaces using Remark 2.2.So let X and Y be two weak P -spaces.Then X * and Y * are P -spaces.Thus X * × Y * X × Y * is a P -space.Therefore, X × Y is a weak P -space.
Note that infinite product of weak P -spaces is not necessarily a weak P -space, since if X is any discrete space containing more than one point and A is infinite; then the product space X A is not a P -space, and, since it is semiregular, it cannot be a weak P -space either.

On Generalized Lindel öf Spaces
One can easily show that if a space is semiregular and nearly Lindel öf or regular and almost Lindel öf , then it is Lindel öf.And it is well known that the Sorgenfrey line E is regular and Lindel öf, but the Sorgenfrey plane E × E is not Lindel öf.Thus, neither of the properties almost Lindel öf and nearly Lindel öf 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 X α is a nonempty topological space.If α∈Δ X α is nearly Lindelöf (resp., almost Lindelöf or weakly Lindelöf), then X α is nearly Lindelöf (resp., almost Lindelöf or weakly Lindelöf).
Proof.Since the projection map π α : X α → X α is a continuous and open function from X α onto X α , it is almost continuous and almost open.Thus π α is θ-continuous and R-map see 20, 21 .Therefore, X α is nearly Lindel öf, almost Lindel öf, and weakly Lindel öf 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 Lindel öf space with a nearly compact space is nearly Lindel öf.Next we prove analogous results concerning almost Lindel öf and weakly Lindel öf spaces.Proposition 3.3.Let X be an almost Lindelöf (weakly Lindelöf) space and Y a nearly compact space.Then X × Y is almost Lindelöf (weakly Lindelöf).
Proof.The proof of Proposition 3.3 is similar to the proof of an analogous result for nearly Lindel öf spaces see 5, Proposition 1.9 .
Note that a space X, τ is nearly Lindel öf if and only if X, τ * is Lindel öf 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 P -space with a nearly Lindelöf space is nearly Lindelöf.
Proof.Let X be a nearly Lindel öf weak P -space and Y nearly Lindel öf.Thus, by Remark 2.2 a , X * is a Lindel öf P -space and Y * is Lindel öf.So, by Proposition 2.1, X * × Y * X × Y * is Lindel öf.Therefore, X × Y is nearly Lindel öf.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 P -spaces, is nearly Lindelöf.
Next we prove that the result in Theorem 3.4 above is correct for almost Lindel öf and weakly Lindel öf spaces.Cl U α yn xm .

3.2
Since the last term is countable, thus X * × Y * X × Y * is also almost Lindel öf and therefore X × Y is almost Lindel öf.
Corollary 3.7.The product of finitely many almost Lindelöf spaces, all but one of which are weak P -spaces, is almost Lindelöf.
For weakly Lindel öf spaces we give the following results.
Theorem 3.8.The product of a weakly Lindelöf weak P -space X with a weakly Lindelöf space Y 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 P -spaces, is weakly Lindelöf.
Next we prove that the product of separable spaces is weakly Lindel öf.First we recall that a space X is called separable if it has a countable dense subset, and one says that X has caliber ℵ 1 if, whenever U is a family of open subsets of X with |U| ℵ 1 , a subfamily V of U exists with |V| ℵ 1 and {V : V ∈ V} / ∅.One also says that X satisfies the countable chain condition if every family of disjoint open subsets of X is countable.Moreover, X is called almost rc-Lindel öf 23 if every regularly closed cover of X has a countable subfamily whose union is dense in X.
Theorem 3.10.The product of separable spaces is weakly Lindelöf.
Note that the product of two separable Lindel öf spaces need not be almost Lindel öf, since the Sorgenfrey line E is separable, but E × E is not almost Lindel öf since it is a regular non-Lindel öf space.
Since in weak P -spaces, for any countable open subsets {U n : n ∈ N} of X, we have Cl n∈N U n n∈N Cl U n .Thus we conclude the following lemma.
Lemma 3.11.In weak P -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 X is a weakly Lindelöf weak P -space and Y is almost Lindelöf, then X × Y is almost Lindelöf.
A ⊆ X is called regularly open regularly closed if A Int Cl A A Cl Int A .The topology generated by regularly open subsets of a space X, τ is called the semiregularization of the space X, τ and is denoted by X, τ * or simply by X * .A space X, τ is said to be semiregular if the regularly open sets form a base for the topology or equivalently τ τ * .By regularly open cover of X we mean a cover of X by regularly open sets in X, τ .Moreover, a space X is called a P -space if every G δ -set is open in X, and it is called nearly compact 13 if every open cover {U α : α ∈ Δ} of X admits a finite subfamily such that X n k 1 Int Cl U k or, equivalently, every regularly open cover of X has a finite subcover.Recall also that a function f from a topological space X to a topological space Y is called R-map 14 almost continuous 15 if f −1 V is regularly open open in X for every regularly open set V in Y .It is called θ-continuous 16 if for every x ∈ X and every open subset V of Y containing f x , there exists an open subset U in X containing x such that f Cl U ⊆ Cl V .Moreover, f is called almost open 15 if f U is open in Y for every regularly open subset U in X, and it is called almost closed 15 if f C is closed in Y for every regularly closed subset C in X.
is an open cover of the Lindel öf space {x} × Y .Thus it has a countable subcollection {{x} × U α x yn : n ∈ N} which covers S x .So the countable family {T α x yn × U α x yn : n ∈ N} covers S x {x} × Y .Now define T x n∈N T α x yn .Since X is a P -space, T x is an open subset in X.Thus {T x × U α x yn : n ∈ N} is a countable open cover of the slab R x T x × Y , and hence all the more {T α x yn × U α x yn : n ∈ N} is a countable open cover of the slab R x .Now the collection of sets {T x : x ∈ X} is an open cover of the Lindel öf space X.So it has a countable set of points {x k Lindel öf, which completes the proof.Remark 2.2. a Recall that a space X is a weak P -space 18 if, for each countable family {U n : n ∈ N} of open sets in X, we have Cl n∈N U n n∈N Cl U n .Clearly, X, τ is a weak P -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 X, τ is a weak P -space if and only if X, τ * is a P -space see Definition 3.1 see 1, 2, 4 .A topological space X is called nearly Lindel öf, almost Lindel öf, and weakly Lindel öf if, for every open cover {U α : α ∈ Δ} of X, there exists a countable subset {α n Theorem 3.6.The product of an almost Lindelöf weak P -space X with an almost Lindelöf space Y is almost Lindelöf.Proof.Since almost Lindel öf property is a semiregular property, that is, a space X, τ is almost Lindel öf if and only if X, τ * is almost Lindel öf see 6, Theorem 2.1 , it is sufficient to prove that X * ×Y * is almost Lindel öf.Thus let {U α : α ∈ Δ} be an open cover of X * ×Y * , and, without loss of generality, suppose that U α V α × W α for every α ∈ Δ where V α is regularly open in X and W α is regularly open in Y .Fix x ∈ X, and, for each y x ∈ Y , there exists α y x ∈ Δ such that x, y x ∈ V α y x × W α y x .Now {W α y x : y x ∈ Y } is an open cover of the almost Lindel öf space Y , so it has a countable subset {W α yn x : n ∈ N} such that Y n∈N Cl W α yn x .Put H x n∈N V α yn x .Since X is a weak P -space, H x is a regularly open set in X.Thus {H x : x ∈ X} is an open cover of the almost Lindel öf space X, so it has a countable subset {H x m : m ∈ N} such that X m∈N Cl H x m .Therefore,