eu FILTER CHARACTERIZATIONS OF PARALINDELÖF SPACES AND META-LINDELÖF SPACES AND SOME CONSEQUENT PRODUCT THEOREMS

Filter characterizations of paralindelöf and meta-Lindelöf spaces are produced. The characterizations are then used to give new proofs of some well-known properties of these spaces. For product spaces, it is also shown that if X and Y are paracompact (respectively, metacompact, paralindelöf, metaLindelöf) then X×Y is paracompact (respectively, metacompact, paralindelöf, meta-Lindelöf), provided that the projection p : X × Y → Y is closed. To Dr. Shashi Prabha Arya. AMS Subject Classification: 54D18, 54C10

etc. using filters or filterbases have been long available in the literature.However, in the cases of significant concepts such as paracompact spaces and metacompact spaces, such characterizations were not available in the literature until recently.This deficiency was overcome when in [5] and [9], respectively, filter and filterbase characterizations for paracompact and metacompact spaces were given.These characterizations were then effectively used to provide new proofs of some established and well-known properties of such spaces.In this article, characterizations of paralindelöf and meta-Lindelöf spaces are developed in terms of filters and filterbases, and new results are found for product spaces using these properties.
Recall that a family of subsets of a topological space is called locally countable (respectively, locally finite) if each point in the space has a neighborhood which intersects at most countably (respectively, finitely) many members of the family.A family of subsets Ω refines a family of sets Γ if for each A ∈ Ω, A ⊆ B for some B ∈ Γ.If a family Ω refines a family Γ, we say that Ω is a refinement of Γ.A space X is paralindelöf (respectively, paracompact) if each open cover Λ of X has a locally countable (respectively, locally finite) open refinement.A space X is meta-Lindelöf (respcetively, metacompact) if every open cover has a point countable (respectively, point finite) refinement.A family of subsets of a space is called point countable (respectively, point finite) if each point in the space belongs to at most countably many (respectively, finitely many) members of the family.If Ω is a filterbase, the adherence of Ω is denoted by adhΩ = ∩ Ω clF .
Studies of paralindelöf spaces and similar spaces are available in [1], [2] and [10].In the second part of [10], P. J. Nyikos points out different open problems in the study of paralindelöf spaces and suggests possible avenues of investigation of these spaces, while recognizing that the study of paralindelöf spaces is a wide open area.In [2], W. G. Fleissner and G. M. Reed give a survey of spaces which are paralindelöf.In the present article, we study the class of paralindlöf spaces and the class of meta-Lindelöf spaces by providing characterizations of paralindelöf spaces and meta-Lindelöf spaces in terms of filters, in a similar fashion as in the case of paracompact spaces [9] and also as in the case of metacompact spaces [5].
It is well-known that every paracompact Hausdorff space is normal.In [2], a series of separation axioms satisfied by paralindelöf spaces are given.In that they assume that all spaces under consideration are regular and T 1 .In particular, they show that every paralindelöf space is pseudonormal, where a space is said to be pseudonormal [2] if every pair of disjoint closed sets, one of which is countable, can be separated by disjoint open sets.It is shown in [2] that a paralondelöf space which is σ-discrete is paracompact, where a space X is called σ-discrete if X = ∪{∪Y n : n ∈ ω} where Y n is a discrete collection of singletons [2].Whether a paralindelöf space is paracompact had been an open question for a long time.In 1982, this question was settled when C. L. Navy [8] gave an example of a paralindelöf T 3 space which was not paracompact.In [11] H. Tamano proved that a space X is paracompact if and only if X × βX is normal.Therefore, as noted by Nyikos [10], by the existence of a paralindelöf space which is not paracompact, there exists a paralindelöf space which is not normal since if X is paralindelöf and not paracompact, then X × βX is not normal, but X × βX is paralindelöf.Throughout this article, all spaces are assumed to be Hausdorff.

Initial Results
We begin this section by defining the following concepts.Definition 2.1.A family Ω of subsets of a space X is said to be locally countably ultimately dominating (l.c.u.d.) if for each x ∈ X there exists an open set U containing x such that U is contained in all but countably many members of Ω.A family Ω of subsets of a space X is said to be countably point dominating (c.p.d.) if each x ∈ X is a member of all but countably many members of Ω.
It is to be noted that if a family F of subsets of a space X is locally countable (respectively point countable), then the family Definition 2.2.A filter Ω on a space is said to be of type PL (respectively, type ML) if each l.c.u.d.(respectively, c.p.d.) subcollection of Ω has non-empty adherence.A base for a filter of type PL (respectively, of type ML) will be referred to as a filterbase of type PL (respectively, a filterbase of type ML).
Before we state our initial theorem, the following Lemma is offered without proof as the proof is straight forward.
Lemma 2.1.A filter Ω is of typeP L (respectively, of typeM L) if and only if every l.c.u.d.(respectively, c.p.d) closed subcollection of the filter has non-empty adherence.
Theorem 2.1.A space is paralindelöf (meta-Lindelöf) if and only if every filter of typeP L(typeM L) on the space has non-empty adherence.
Proof.Suppose that X is paralindelöf and that Ω is a filter on X such that adhΩ = ∅.Then F = {X − clF : F ∈ Ω} is an open cover of the paralindelöf space X.Hence there is a locally countable open refinement K for F. Let and is a cover of X.Therefore, Ω is a filter of not typeP L. Hence every filter of typeP L has non-empty adherence.
Conversely, let every filter of typeP L on X has non-empty adherence.Suppose that Ω is an open cover of X which has no locally countable refinement.Therefore, Ω has no countable subcover.Consider U = {X − ∪ Γ F : F ∈ Ω, and Γ is countable}.Then U is a base for a filter on X with empty adherence because, Consequently, in view of Lemma 2.1, there is a closed, l.c.u.d.subcollection Λ, of the filter generated by U , with empty adherence.Then F = {X − F : F ∈ Λ} is locally countable, since Λ is a l.c.u.d.family.Also, ∪ Λ (X − F ) = X.Since Λ is a subcollection of the filter generated by U , for each In view of the Theorem 2.1 and Lemma 2.1, the following Theorem is immediate.
Theorem 2.2.A space is paralindelöf if and only if every closed filter of typeP L on the space has non-empty adherence.
The characterizations for meta-Lindelöf spaces can be obtained by replacing the filter of typeP L with filter of typeM L. The proof will follow along the same line of argument and construction.
Theorem 2.3.A space X is meta-Lindelöf if and only if every filter (closed filter) of typeM L has non-empty adherence.
We shall use the above characterization and the following Lemma 2.2 to give new proofs of some properties of paralindelöf and meta-lindelöf spaces and continuous images of them, for example, as can be seen in the following theorems.
Lemma 2.2.If f : X → Y is continuous and Ω is a filter of typeP L(typeM L) on X, then the filter generated by {f (F ) : F ∈ Ω} is a filter of typeP L(typeM L) on Y .
Proof.Let Γ be a closed l.c.u.d.subcollection of the filter generated by {f (F ) : Following is an easy consequence of Lemma 2.2.Lemma 2.3.If X is a space and A ⊆ B ⊆ X, then any filter of typeP L (typeM L) on A is a filter of typeP L(typeM L) on B.
Proof.The proof is clear, in view of Lemma 2.2, since the identity function from A to B is continuous.
Note that on a space X, a filter Ω of type PL(type ML) has non-empty adherence if and only if a base B of Ω has non-empty adherence.In view of this fact, Theorem 2.1 and Theorem 2.2 above, the following characterization follows: Theorem 2.4.A space X is paralindelöf (respectively, meta-Lindelöf) if and only if every filterbase (closed filterbase) of type PL(respectively, type ML) has non-empty adherence.
Proof.The proof is straight forward, in view of Lemma 2.3 Theorem 2.6.Each subspace of a paralindelöf (respectively, meta-Lindelöf) space is paralindelöf (respectively, meta-Lindelöf) if and only if each open subspace is paralindelöf (respectively, meta-Lindelöf).
Proof.If each subspace is paralindelöf (respectively, meta-Lindelöf), then each open subspace is paralindelöf (respectively, meta-Lindelöf).Conversely, let each open subspace of a space X is paralindelöf (respectively, meta-Lindelöf).Suppose that A is a subspace of X and that Ω is a filterbase of typeP L(typeM L) on A such that A ∩ adhΩ = ∅.Then A ⊆ X − adhΩ and Ω is a filterbase of typeP L(typeM L) on X − adhΩ.Also, (X − adhΩ) Proof.To show that X is paralindelöf, in view of Theorem 2.4, we shall show that each closed filterbase of typeP L on X has non-empty adherence.Suppose that Ω is a closed filterbase of typeP L on X.Then {f (F ) : F ∈ Ω} is a closed filterbase of typeP L on the paralindelöf space Y .Therefore, there exists Proof.The projection p : X × Y → Y is continuous and closed.Also for each w ∈ Y, p −1 (w) is homeomorphic to the compact space X.Hence the result follows from Theorem 2.7.
Remark.Each of the above results is true, if 'paralindelöf' is replaced by 'meta-Lindelöf' and the filter (filterbase) of typeL be replaced by filter (filterbase) of typeM L.
Proof.Follows from Theorem 2.7 since a Lindlöf space is paralindlöf.
Proof.Follows from Theorem 2.7 above and Theorem 4 of [9], using a similar argument as in the proof of Corollary 1, since a regular Lindelöf space is paracompact.(Corollary 1 of [9]) Corollary 2.4.Let X be regular and f : X → Y be a closed, continuous and onto function with f −1 (y) compact for each y ∈ Y .Then X is paralindelöf if and only if Y is paralindelöf.
Proof.Follows from Theorem 2.7 above and the Corollary 3.2 of [1] which states that if X is paralindelöf and f : X → Y is a continuous closed onto function with f −1 (y) Lindelöf for each y ∈ Y , then Y is paralindelöf.
Lemma 2.4.Let Ω be a filterbase of typeP L with the property that countable intersection of members of Ω belongs to Ω and let U = {U |U = ∩ Γ F, F ∈ Ω, Γcountable }.Then U is a filterbase of typeP L.
Proof.Consider that Ω is a filterbase of typeP L with the indicated property and let U = {U |U = ∩ Γ F, F ∈ Ω, Γ countable}.Suppose that B is a l.c.u.d.subcollection of the filter generated by U .Then B is a subcollection of the filter generated by Ω, since for each B ∈ B, there is a countable subfamily Γ such that U = ∩ Γ F ⊆ B, F ∈ Ω.Since ∩ Γ F ∈ Ω, Γ countable, this implies that B is a subcollection of the filter generated by Ω and hence has an adherent point.Hence U is a filterbase of typeP L. Lemma 2.5.If f : X → Y is continuous and if Ω is a filterbase of typeP L on X with the property that countable intersections of members of Ω belong to Ω, then {f (F ) : F ∈ Ω} is a filterbase of typeP L on Y with property that countable intersections of members of {f (F )} belong to {f (F )}.
Proof.In view of Lemmas 2.2 and 2.4, if Ω is a filterbase of typeP L on X, then Then W is a family of closed subsets of f −1 (w) with the countable intersection property and f −1 (w) is Lindelöf.Therefore, there is a p ∈ f −1 (w) such that p ∈ ∩ W W ⊂ ∩ Ω F .Hence adhΩ = ∅.Hence X is paralindelöf.
Corollary 2.5.Let X be regular and f : X → Y be closed continuous onto function with f −1 (y) Lindelöf for each y ∈ Y .Then X is paralindelöf if and only if Y is paralindelöf.
Proof.Follows from Theorem 2.8 above and the Corollary 3.2 of [1].
Note that the Corollary 2.5 improves on the Corollary 2.4, since every compact space is Lindelöf.

Product Theorems
In this section we shall use the characterizations of paracompact spaces, metacompact spaces, paralindelöf spaces and meta-Lindelöf spaces in terms of filters introduced here and in [5] and [9] to prove that the product of these spaces preserves the property, provided that the projection satisfies a closedness condition.
In [5] and [9] respectively, characterizations for metacompact spaces and paracompact spaces are given using filters and filterbases.It is shown in [9] that a space X is paracompact if and only if every filterbase (or closed filterbase) of typeP has non-empty adherence.A filterbase Ω is defined to be of typeP [9] if each locally ultimately dominating (l.u.d.) filter subbase coarser than Ω has non-empty adherence.A family is locally ultimately dominating (l.u.d) if for each x ∈ X there is an open set about x contained in all but finitely many elements of Ω.A space X is metacompact [5] if and only if every filter (or closed filter) of typeM on X has non-empty adherence.A filter on a space is of typeM [5] if every point dominating (p.d.) subcollection of the filter has nonempty adherence.A collection Ω of subsets of a space is point dominating (p.d.) if each x ∈ X is a member of all but finitely many members of Ω.
Note that the characterization of paracompact spaces given in [9] in terms of filter bases can be easily stated in terms of filters of type P, where a filter Ω is defined to be of type P if every l.u.d.subcollection of Ω has a non-empty adherence.The proof of Theorem 3.1 can essentially be done as the proof of Theorem 1 of [9].
Theorem 3.1.A space X is paracompact if and only if every filter of type P on the space has non-empty adherence.