Some thoughts on countable Lindelöf products

Topology and its Applications 259 (2019) 287–310

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Some thoughts on countable Lindelöf products N. Noble P.O. Box 1812, Los Gatos, CA 95031, United States of America

a r t i c l e

i n f o

Article history: Received 3 April 2018 Received in revised form 18 August 2018 Accepted 23 August 2018 Available online 15 March 2019

a b s t r a c t A space X for which Xω is Lindelöf is called powerfully Lindelöf. Extending this term, I call a collection of topological spaces powerfully Lindelöf if the product of each of its countable subcollections is Lindelöf. In 1971 E. Michael showed (CH) that there is no single such class; here I explore what can be said of them. In doing so, I will focus upon some of the techniques used to prove countable products Lindelöf. © 2019 Published by Elsevier B.V.

MSC: primary 54B10, 54D20 secondary 54A10, 54Cxx, 54E15, 54G10, 54G12, 54G99, 54H05 Keywords: Lindelöf products Productively Lindelöf Powerfully Lindelöf Totally Lindelöf Alster

0. Introduction

In 2010 I received an email from Wis Comfort asking for comments on an attached draft of a paper, Comfort [17], which he was writing to commemorate Mel Henriksen, specifically drawing my attention to a section exploring a technique, introduced by Henriksen, Isbell, and Johnson [34], for showing spaces to be Lindelöf. He prefaced that request with a comment that it “fell into my area of expertise”. Nothing unusual among colleagues, but a little surprising in my case because I had left mathematics about forty years earlier. Before that, circa 1969-70, I had indeed been interested in countable Lindelöf products - they were “in the air”: Tony Hager [31] had recently publicized the fact that countable products of σ-compact and Lindelöf Čech complete spaces were Lindelöf, giving concise proofs based upon the Henriksen-Isbell-Johnson technique mentioned above; I (answering a question posed by Hager) had stumbled upon the fact that countable products of Lindelöf P-spaces were Lindelöf; and Arthur Stone, Alexander Arhangel’skii, Ernie

E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.02.037 0166-8641/© 2019 Published by Elsevier B.V.

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Michael, and many others, exploring paracompactness and other properties related to metrizability, were turning up other interesting powerfully Lindelöf spaces. In our relationship Wis always gently conveyed the view that just because someone had fallen from the path of mathematics didn’t mean that they could not repent and return, and I suspect that part of his motivation in including me among the readers of his draft was to see if it might rekindle my interest in abstract mathematics; if so, he succeeded. Somewhat surprisingly, I was actually able to make a useful suggestion regarding his treatment of Lindelöf products and in the process was drawn back to topology in a mainly passive way - I began browsing the point-set topology literature, looking into topics which had once held my interest. (And looking at papers which referenced N. Noble - what can I say?) Thus “prepared”, when the invitation to contribute to this Wis Comfort Special Issue arrived I decided to attempt to do so, and in fact to return to the subject of countable Lindelöf products. In modern terminology, the initial question had been: Is there a single powerfully Lindelöf class? That is, did Xω , Yω Lindelöf imply (XxY)ω Lindelöf? Already in ’69-70 Ernie had answered that question in the negative, (assuming CH), with his remarkable collection Michael [43] of examples sketching the boundaries of Lindelöf product theory. But those results only served to bring to the fore the less precise “What do the powerfully Lindelöf classes look like?” As I began reviewing the literature I found myself focusing less upon the particular classes and more upon the technique used in the proofs, both as an approach to unifying the results in my mind and as a way of avoiding the tedious (and in my case, error prone) task of considering generalizations to larger cardinals. If the structure of the proof is adequately exposed, those with a need for “implied” higher cardinal analogs will (it is my excuse) easily prove them. Thus my goal became more one of exploring the techniques used to prove countable products Lindelöf. With a nine month time window (and substantial knowledge deficit) I have also allowed myself the luxury of barely introducing some topics, thus presenting them as opportunities for future investigation. I would like to thank Tom Peters for several suggestions which have greatly improved my exposition of these thoughts. 1. Research related to countable Lindelöf products I use terms like “finite product” and “countable product” to describe the product over a set of factors with the indicated cardinality and describe a class as closed under finite or countable products with the obvious intent. Infinite Lindelöf products have only countably many non-trivial factors (“trivial”, in this case, meaning compact) and must have each finite subproduct Lindelöf. Furthermore, factors of countable Lindelöf products need not be finitely productive; consequently many countable Lindelöf product theorems start with the premise that the finite subproducts are Lindelöf. Thus any broad consideration of infinite Lindelöf products will involve consideration as well of finite products. A space X is called productively Lindelöf (Barr, Kennison, and Raphael [12] where also the term “powerfully Lindelöf” was introduced) if each product of X with a Lindelöf space Y is Lindelöf. Whether all productively Lindelöf spaces are powerfully Lindelöf spaces is not known, but powerfully Lindelöf spaces need not (at least in some set theories) be productively Lindelöf: ω ω (the irrationals) is powerfully Lindelöf but, as Michael’s example [43, 1.3] showed, there can exist (CH) a Michael space whose product with ω ω is not Lindelöf. Some of the results mentioned here were first presented for spaces satisfying particular separation assumptions which may have since been relaxed or removed. Dealing with finite powers of a single space, Gerlits and Nagy [26] give an internal characterization of those X such that Xn is Lindelöf for all n. An earlier characterization of such spaces X had been given by Arhangel’skii and Pytkeiev [7] in terms of countable tightness of their function spaces. Second countability is preserved by countable products and thus so too are the properties of being a Lindelöf p-space (perfect preimage of a second countable space) or a Lindelöf Σ-space (continuous image of

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a closed subspace of a p-space). Internal characterizations of Lindelöf p-spaces and Lindelöf Σ-space were given by Arhangel’skii [5] and Nagami [45] respectively. Both p-spaces and Σ-spaces, as well as several related properties (starting with spaces admitting a development) which have been studied in the field of generalized metrizability have been characterized in terms of behavior involving countable collections of covers - see Arhangel’skii [6]. Beyond metrizability, Čech complete spaces (a Gδ in its Čech-Stone compactification) have been characterized as complete with respect to a countable collection of open coverings [23,4] and Lindelöf Čech completeness is preserved by countable products [23,21]. Using a quite general concept of completeness, Frolik [24] considered three classes of powerfully Lindelöf spaces: a class now known as Frolik spaces (the closed subspaces of countable products of σ-compact Tychonoff spaces); a class he called “B-spaces”, and analytic spaces (continuous images of the irrationals). He gave internal characterizations of each class in terms of completeness with respect to countable collections of, respectively, countable closed covers, countable covers, and a certain highly structured class of countable covers. Frolik’s work with analytic spaces has been continued, e.g. Miller and Fremlin [44], Tall [61], with the exploration of K-analytic spaces, the class generated as continuous images of Lindelöf Čech complete spaces. Hager, [31], provided a concise proof that countable products of Lindelöf Čech complete spaces and of Hausdorff σ-compact spaces were Lindelöf using the distinguished subspace technique mentioned earlier and Comfort [17] expanded upon those results, showing that the class of spaces distinguished by closed compact subspaces was closed under countable unions, countable intersections, and countable products. Lindelöf P-spaces (each Gδ open) are productively Lindelöf and preserved by finite products, but not by countable products. However, their countable products are Lindelöf, Noble [48, Corollary 4.2]. In [66] Vaughan introduced totally Lindelöf spaces, a class which includes both Lindelöf P-spaces and σ-compact spaces and showed Vaughan [67] that they are productively Lindelöf, preserved by finite products, and have Lindelöf countable products. Vaughan also showed that a totally Lindelöf space in which each compact subset is contained in a σ-compact Gδ (for example, any totally Lindelöf metrizable space) is σ-compact. Alster [3] introduced a finitely productive productively Lindelöf powerfully Lindelöf class of spaces, now known as Alster spaces, which also contain both Lindelöf P-spaces and σ-compact spaces and have the property that they are σ-compact if each of their compact subsets is contained in a σ-compact Gδ . Alster also showed (CH) that each productively Lindelöf space of weight not greater than ℵ1 is Alster. This result has since been improved and extended in various ways - for example, by showing that (CH) each productively Lindelöf T3 space of cardinality ℵ1 is powerfully Lindelöf - Tall and Tsaban [62, Corollary 1.5]; that, (CH), all productively Lindelöf spaces which are the union of ℵ1 compact sets or are of countable tightness are powerfully Lindelöf, Burton and Tall [14] and Medini and Zdomskyy [42] respectively; and by results weakening the assumption of CH. Alster spaces were discovered independently (of Alster) by Barr, Kennison, and Raphael [12] and were given the name “Alster” in Barr, Kennison, and Raphael [13], where a perhaps different class, OCR-scattered spaces, was also studied. The OCR-scattered spaces are Alster and include the continuous images of Lindelöf P-spaces (each Gδ cover has a countable subcover) and thus the Lindelöf SP-scattered spaces studied by Henriksen, Raphael, and Woods [35]. In a parallel result, I show in Section 5 that regular totally Lindelöf spaces are Alster. Since 2009 there has been a vigorous effort by Franklin D. Tall and a group of colleagues and coauthors to understand Alster and related spaces and their relationship to the class of productively Lindelöf spaces. Their findings include the countable tightness result of Medini and Zdomskyy, mentioned above, and a continuing exploration of the relationships among Lindelöf and related properties including Hurewicz, Menger, “D”, and indestructibly productively Lindelöf: Scheepers and Tall [54]; Tall [57], [58], [60] and [61]; and Duanmu, Tall, and Zdomskyy [20]. A striking recent paper from two members of this group, Aurichi and Zdomskyy [9], gives an internal characterization of productively Lindelöf spaces! While the definitions of totally Lindelöf, Alster, or OCR-

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scattered spaces may provide such a characterization, they are not known to do so. In the same paper Aurichi and Zdomskyy enhance the list of possible characterizations by introducing “weakly Alster spaces”, a class which includes (and perhaps equals) the class of Alster spaces and is included in (and perhaps equal to) the class of productively Lindelöf spaces. Michael [43], adapting a proof used by Okuyama [50] (perhaps due to Morita) showed that a countable product was Lindelöf provided each of its finite subproducts had the property that each closed subset was a Gδ . Such spaces are hereditarily Lindelöf, and in fact essentially the same proof shows that a countable product is hereditarily Lindelöf provided this is true of each of its finite subproducts, a fact which does not seem to appear in the literature until Zenor’s more general treatment of “m-Lindelöf, m-discrete” spaces Zenor [70]. Proofs similar to Okuyama’s were used by Nagami [46] to show that a countable product with each finite subproduct Lindelöf is itself Lindelöf provided it is countably paracompact, and by Zenor [69] to show that a countable product with each finite subproduct normal is itself normal if and only if it is countably paracompact (implying the corresponding result for regular Lindelöf spaces). Michael [43] also includes a collection of examples sketching the boundaries of Lindelöf product theory, including (CH) the original example of a Michael space. 2. Conventions, notation, and a preliminary result I make no hidden separation assumptions; in particular, compact spaces need not be Hausdorff, Lindelöf spaces need not be regular, and neither regular nor sober spaces need be T0. I take T3 as being regular T1 . Note that Alexander’s Subbase Lemma, Tychonoff’s Product Theorem, and Wallace’s Theorem are valid where “compact” simply means “each open cover has a finite subcover”. I will be a bit cavalier in how I start proofs verifying the Lindelöf property. Since a countable subcover of a refinement identifies a countable subcover of the cover refined, it suffices to consider only covers made up of basic open sets. In the other direction, it suffices only to consider covers expanded to include countable unions (since, given a countable subcover of such an expansion, each such union can be replaced by its components to provide a countable subcover of the original cover). And, instead of considering a cover, it suffices, given a collection of open subsets closed under the formation of countable unions which does not contain the space in question, to demonstrate that that collection can not be a cover. I will usually start a proof with which ever type of collection is convenient, without further allusion to its consideration being sufficient.   Products X = α Xα (sometimes identified more explicitly as α∈A Xα ) are assumed to be indexed by a set A well ordered by the least ordinal of cardinality |A|. Our focus is products with at most countably many factors which we will, in the infinite case, typically represent with the format Xω if the factors are   identical or X = n Xn (= n∈N Xn ) if they may differ. Use of the index “i” will indicate a finite product:   i Xi = i, in subscripts and superscripts, It will be convenient to use the relational symbols ∈, ∈, with the understanding that X<0 = 1, the one point space, and other usage as indicated by these examples:  / • α∈A Xα = Xα∈F xXα∈A\F = Xα∈F xXα∈F ;   n ; • πn X = Xn , π≤n X = X≤n ; X = X≤n x π>n X. I will also use the “closure bar” with π: for example, π
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 U containing S there exists an open product set V = α∈A Vα (where each Vα is open and Vα = Xα with at most finitely many exceptions) with S ⊆ V ⊆ U. Also, for S a class of spaces call the product space X  S-Wallace if each product set α Sα drawn from S is a Wallace subspace of X. P-spaces are finitely productive: this follows immediately from the observation that if each countable intersection of basic open subsets is open then so is each countable intersection of open sets. Lindelöf P-spaces are productively Lindelöf (and thus finitely productive) and the standard proof of that fact (indeed, of the corresponding proof for [α, ∞]-compact space <α-discrete spaces) adapts to give Wallace subspace results. I state only the Lindelöf case: 2.1 Proposition. A finite product of Lindelöf P-spaces is Wallace. Proof. It suffices to consider the product of two such subspaces. Suppose U is an open set containing SxT with S and T (and SxT) Lindelöf P-spaces and let U be a cover of SxT by open product sets contained in U. For each s in S there exists (since T is Lindelöf) a subset {U(s, n)xV(s, n)} of U which covers {s}xT. Set   Us = n U(s, n) and Vs = n V(s, n), and note that {s}xT ⊆ Us xVs ⊆ U. The sets Us are open (since S is a P-space), and thus the cover {Us } has a countable subcover (since S is Lindelöf), say {Wm }. Label as Vm   the set Vs corresponding to Wm = Us ; set V = m Vm ; and set W = m Wm . Then V is open (since T is a P-space) and SxT ⊆ WxV ⊆ U, so SxT is Wallace. 3. Distinguished families I: compact subsets A subspace X of a space Y is distinguished by a family of subspaces D if for each x ∈ X and each y ∈ Y\X there exists D ∈ D with x ∈ D and y ∈ / D. If for each α ∈ A a family Dα distinguishes Xα ,  α=β then {Dβ xX : Dβ ∈ Dβ , α, β ∈ A} distinguishes α Xα (from the product of the host spaces). Thus properties characterized in terms of distinguishing families are candidates for the “X has P if each factor has P” format, provided their characterization does not introduce conflicting factors. Not having been specified, the host for a space distinguished by a countable family of subsets might be any space, but since any space (Y, τ ) can be embedded in its Alexandroff (one-point) compactification   (Y∗ , τ ∗ ) where Y∗ = Y {y∗ }, y∗ ∈ / Y, and τ ∗ = τ {U ⊆ Y∗ : Y∗ \U ⊆ Y is closed and compact}, we may suppose the host is compact. As we discuss presently, spaces distinguished by a “suitable” countable family of compact subsets are Lindelöf. Families of closed compact sets are suitable in this sense and we will explore more general conditions meeting this requirement. The concept of distinguished subspaces was introduced by Henriksen, Isbell, and Johnson [34, Lemma 2.2], who used it to identify Lindelöf subspaces of a compact Hausdorff space. It was subsequently used by Hager [31] to prove Lindelöf countable products of σ-compact spaces and countable products of Čech complete spaces. The distinguished subspaces approach has been generalized to higher cardinalities Ball, Hager, and Neville [11, 4.6] and Ball and Hager [10, 1.4(iv)-(vi)]. Other examples of its use can be found in the monographs Comfort and Negrepontis [18] and Engelking [22, 3.8.F], and as previously mentioned, Comfort [17] where distinguished subspaces are considered in detail. These applications of the distinguished subspace approach to product theory involve collections of compact subspaces which enjoy Hausdorff separation or a work around - a requirement that they be closed. In extending this approach to a wider context we will need a slightly different concept. For X a subspace of Y and D a collection of subsets of Y, call D focused at X if X is nonempty and for each x ∈ X and any open U containing X there exists Dx in Dx = {D ∈ D : x ∈ D} with Dx ⊆ U. Notice that while “D distinguishes  X” ensures that Dx ⊆ X, “D focused at X” only implies the intersection is contained in the saturation of X (the intersection of the open sets which contain X). However, a subspace and its saturation share the same open covers and thus the same Lindelöf number, so in many cases “focused” will, for our purposes, suffice. Plus, in the most important cases, “distinguished” implies “focused”.

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Recall that a collection of subsets is filtered if each nonempty intersection of finitely many of its members contains a nonempty member, and call it δ-filtered if each such intersection of countably many members contains a nonempty member. A family closed under the formation of finite intersections is filtered and a filtered family in which each finite intersection is nonempty is a filter base. Call a family in which each countable intersection is nonempty δ-stable. An important, well studied, and apparently unnamed property of some filtered families, which I call tight intersections, is: if some nonempty intersection of members is contained in an open set, then some nonempty member of the family is contained in that open set. Filtered families of closed compact subsets have tight intersections, and it is the topologically significant aspect of the Hofmann-Mislove Theorem that filtered families of saturated compact subspaces of a T0 sober space have tight intersections. (A subspace is saturated if it equals its saturation.) 3.1 Remark. If a filtered family with tight intersections distinguishes X, it is focused at X.  Proof. For D such a family, U ⊇ X open, x ∈ X, and Dx = {D ∈ D : x ∈ D}, Dx ⊆ X ⊆ U because D distinguishes X and hence, since D has tight intersections, there exists Dx ∈ Dx with Dx ⊆ U, as desired. Recall that a filter base F is said to converge to a set X if X ⊆ U, U open implies ∃F ∈ F with F ⊆ U. If D is focused at X, then the collection of covers of X drawn from D converges to X. This need not be the case if D only distinguishes X and is the reason I have introduced the “focused” condition. 3.2 Proposition. If D is focused at X and Φ is the collection of unions of covers of X drawn from D, then Φ converge to X. Consequently, if each union in Φ contains one which is Lindelöf, X is Lindelöf. Proof. Suppose U is an open set containing X. Since D is focused there exists, for each x ∈ X, a set Dx ∈ D  with x ∈ Dx ⊆ U. For C = x Dx , C is in Φ and satisfies X ⊆ C ⊆ U, and therefore Φ converge to X. To see that X is Lindelöf provided the members of Φ are as indicated, suppose U is an open cover of X, let  U = U, and let C be a Lindelöf member of Φ with X ⊆ C ⊆ U. Since C is Lindelöf there exists a countable subcollection of U which covers C and therefore, as desired, X. For D a collection of Lindelöf spaces, the requirement that each union in Φ contain one which is Lindelöf is satisfied if D is countable or if each cover of X drawn from D has a countable subcover, in particular if each member of D actually contains X. The requirement that Φ derive from all covers drawn from D can  be relaxed so long as, in the proof, each Dx can be found such that C = x Dx is contained in a member of Φ. Our next result considers collection of spaces each the focus of a suitable collection. 3.3 Proposition. Suppose that for each n Yn is compact and Xn ⊆ Yn is associated with a countable collection     of subspaces Dn . Set D = { n∈B Dn x n∈B / Yn , Dn ∈ Dn , B finite} and D = n Dn ,  (a) For X = n Xn , if each Dn is focused at Xn , then D is focused at X provided  • products n En , En ∈ Dn , are Wallace and each Dn is filtered, or • each Dn is δ-filtered.  (b) For X = n Xn , if each Dn is focused at Xn , then D is focused at X.  (c) For X = n Xn , if each Xn is distinguished by Dn , then D distinguishes X, and   (d) identifying X = n Xn with the diagonal of n Xn , D distinguishes X. Proof. Let x ∈ X and let U be an open set which contains X.  (a). Write the countable collection Dx = {D ∈ D : x ∈ D} of product sets as {Dn = n Dnm } where,    each Dn being filtered, we may assume Dnk ⊆ Dnm for k > m. Note that E = n Dn = n m Dmn is  contained in U. If E is Wallace, then there exists an open product neighborhood V = j
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E ⊆ V ⊆ U. Since each Dj is focused there exists, for each j < n, m(j) such that Djm(j) ⊆ Vj . Thus for m  sufficiently large, D = j subspaces with empty intersection, so some <finite, countable> subfamily has empty intersection, i.e., the intersection of such a subcollection of C is contained in U. Since D is <filtered, δ-filtered>, it follows that some nonempty member of D is contained in U, as desired. (c). We may (and do) assume that C includes a sober member Y. Suppose U does not contain a member of D and consider the family Φ of covers of C by open subsets of Y which contain no member of D. Since

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 {U Y} is a member, Φ is nonempty, and since members of D are compact, membership in Φ is a property of finite character. Thus by the Techmüller-Tukey Lemma the collection Φ has a maximal member; let M be the union of such a member and let I = Y\M. The closed set I is irreducible: if V and W were disjoint open   sets each of which met I, then by the maximality of M both M V and M W would contain members of the filtered family C and hence M would contain a member of C. Since Y is sober, I is the closure of some point x. Consider any set D in C. Any open set containing D meets I and thus contains x, so x is in each open set  containing D and hence, since D is saturated, x is in D. Since this is true for each D in C, x is in C = C and hence x ∈ M. But x ∈ I = Y\M; this contradiction shows that U must contain a member of C and hence that D has tight intersections. Examples (a) and (b) are well known and (c) is, for T0 spaces, part of the Hofmann-Mislove Theorem. Poncet [52] explores the identification of filtered families of compact subsets with tight intersections in spaces not necessarily T0 . This proof is based on that of Keimel and Paseka [38]. Notice that each saturated subspace of a sober space is sober (Poncet [52, Lemma 3.4]), so in essence (c) deals with families of compact saturated sober subspaces.   For (X, τ ) any T1 space let Y = X {a, b} with topology τ {U : Y\U is finite}. Y is compact T1 , and   X {a} and X {b} are compact (saturated) subsets whose intersection, X, need not be compact. A space in which the intersection of any two saturated compact subsets is compact (and of course saturated, making it easy to find filtered families of saturated compact subsets) is said to be coherent. Such spaces are studied, for example, in the monographs Gierz, Hofmann, Keimel, Lawson, Mislov, and Scott, [27] and [28]; and Goubault-Larrecq [30]. The results from the literature plus those discussed above exhibit the utilization of distinguishing families as a technique for identifying and classifying Lindelöf spaces. The potential reach of such a approach is suggested by examples from the well studied case of Tychonoff spaces. 3.6 Examples of focused families. Suppose D is a class of subspaces of βX and let P = P(D) be the class of spaces X which, (a) through (e), are distinguished by such a D and, (f) and (g), are the focus of such a D. Examples of the pairings of P and D include: (a) (b) (c) (d) (e) (f) (g)

P = compact; P = σ-compact; P = Frolik; P = {identified by Comfort}: P = Lindelöf Čech complete: P = Hurewicz: P = Lindelöf T3 :

D = a closed subset of βX; D = an Fσ subset of βX; D = an Fσδ subset of βX; D = a space in the σ-lattice generated by closed subsets of βX; D = a countable collection of Fσ subsets F with X ⊆ int(F) ⊆ βX; DG = {Fσ subsets F: X ⊆ F ⊆ G} for each Gδ G with X ⊆ G ⊆ βX; D = {Fσ subsets F: X ⊆ F ⊆ βX}.

Proof. Parts (a) and (b) are trivial and part (c), due to Frolik, will be discussed in the next Section. Part (d) is a special case of Comfort [17, Theorem 2.6], which is stated for the σ-lattice generated by closed compact subsets of any space. For part (e), where D distinguishes X, D={Fn } with X ⊆ int(Fn ) ⊆ βX implies that X is a Gδ in βX, i.e., that X is Čech complete. Because each neighborhood of X contains an Fn which, being an Fσ is Lindelöf, X  is Lindelöf. Conversely, where X = n Un with each Un open, cover X for each n with closed neighborhoods  contained in Un . Choose a countable subcollection {Fmn } whose interiors cover X, and set Fn = m Fmn . Then D = {Fn } is as desired. Part (f): Tall [59, Theorem 6] characterizes a Lindelöf T3 space X as Hurewicz if and only if every Čech complete space containing X contains a σ-compact subset which contains X. Thus for X Hurewicz each DG

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is focused at X (for U⊆ X open, DU  G and hence DG contains an Fσ subset which contains X). In the other direction suppose X is the focus of each DG and G ⊇ X is Čech complete, that is, a Gδ in βG. For f : βX → βG the continuous extension of the identity on X, H = f −1 (G) is a Gδ and thus contains, since DH is focused at X, an Fσ subset F. Then f(F) is an Fσ and X ⊆ f(F) ⊆ G, as desired. Part (g): if X is the focus of Fσ subsets of βX each neighborhood of X contains a Lindelöf (indeed, σ-compact) subspace so (by 3.2) X is Lindelöf. Conversely, as above, if X is Lindelöf each neighborhood of X contains an Fσ which contains X so X is the focus of D. In fact, as we note the next section where spaces complete in a very general sense will be considered, a result by Frolik shows that any “complete” normal T1 space X can be distinguished by a family of closed subsets of βX. This includes, of course, all “complete” Lindelöf T3 spaces. There is a wide literature correlating properties of X with its situation in βX, in particular with properties of its remainder βX\X. Among these spaces Lindelöf at infinity (βX\X Lindelöf) which generalize Čech completeness are perhaps of interest in our context, but are not explored here. 4. Distinguished families 2: complete subsets Loosely following Frolik [24], call a family Φ of collections of subsets of a set X a completion structure and define a completion space to be a pair (X, Φ) where Φ is a completion structure. If also (X, τ ) is a topological space, then call (X, τ , Φ) a topological completion space. A family of nonempty subsets F of (X, Φ) is Cauchy if it is filtered and for each ϕ ∈ Φ ∃P∈ ϕ and F∈ F with F ⊆ P; and (X, τ , Φ) is complete  if F = ∅ for each Cauchy family F, where F = {F : F ∈ F}, closure taken with respect to τ . Frolik’s completeness generalizes the more familiar notion of completeness with respect to a uniformity, but retains sufficient power to be of interest. Frolik established several basic properties of spaces complete in this sense, which I list below. 4.1 Facts about spaces complete in the sense of Frolik. If (X, Φ) and, for each n, (Xn , Φn ) is complete, then for the structures (Y, Ψ) listed below, (Y, Ψ) is complete. (a) (b) (c) (d) (e) (f)

X⊂Y and Ψ = Φ. Y = X and Ψ “refines” Φ, that is, each ϕ ∈ Φ is refined by an ψ ∈ Ψ.  Y = X and Ψ = { Φ : Φ ⊆ Φ is finite}. Ψ = {f(ϕ) : ϕ ∈ Φ}, where f : X → Y is continuous and injective. Ψ = {f −1 (ϕ) : ϕ ∈ Φ}, where f : Y → X is perfect (including being surjective).  Y = n Xn and Ψ = {{πn−1 (ϕ) : ϕ ∈ Φn } : n ∈ N}.

Source. Frolik [24]: (a); Theorem 2 ; (b) Proposition 3.2; (c) Theorem 1; (d) Theorem 2; (e) Theorem 3; (f) Proposition 4.7. After developing some general theory Frolik focused his attention on spaces complete with respect to sequences of countable coverings, specifically: • E-spaces, defined as T3 spaces complete with respect to a sequence of closed countable covers (E-spaces have since, Burton and Tall [14], been renamed Frolik spaces); • B-spaces, defined as Hausdorff spaces for which there exists a sequence of countable covers with respect to which they are complete; and • analytic spaces, defined as the spaces which are T3 continuous images of E-spaces.

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Noting that the spaces in each of these classes are Lindelöf, he characterized E-spaces/Frolik spaces as closed subspaces of countable products of T3 σ-compact spaces and also (as referenced above) as those spaces which are an Fσδ in their Čech-Stone compactification, and showed that each Frolik space is the continuous image of a Čech complete Frolik space. Using some of the facts listed above Frolik noted that the property of being a Frolik space is inherited by closed subspaces and preserved under T3 perfect preimages, finite unions, countable intersections, and countable products. Similarly, the property of being a B-space is inherited by closed subspaces and preserved under continuous 1-1 images, Hausdorff perfect preimages, countable products, and, if T3 , countable unions and countable intersections. For Tychonoff analytic spaces Frolik developed a characterization of them as complete with respect to a sequence of closed covers which are subject to certain additional constraints and showed that for metric spaces this definition of analytic (as a continuous image of a Frolik space) coincides with “analytic” in the classical sense - a continuous image of the space of irrational numbers in the unit interval. He then noted that his property of being analytic is inherited by closed subspaces and preserved under T3 continuous images and countable products. To put these properties in context, we mention that Frolik (and independently Arhangel’skii) characterized Čech complete spaces as complete with respect a countable collection of open covers and that realcompactness is completeness with respect to {{f −1 ([0, n]) : f ∈ C(X)} : n ∈ N}, the zero set uniformity. More specialized countable sequences of covers have been studied and used in the exploration of metrizability and related properties, starting with Moore’s introduction in 1910 of developments. As a particularly pertinent example, Lindelöf Σ-spaces (continuous images of closed subspaces of Lindelöf p-spaces, Lindelöf p-spaces being the perfect preimages of second countable spaces - some separation may apply) can be characterized as those complete, in a certain sense, with respect to a countable collection of closed locally finite covers. Sequences of open covers also appear in the selection principles related to game theory and have given rise to Hurewicz and Menger spaces whose relationship to productively Lindelöf spaces has seen extensive study. See, inter alia, Alas, Aurichi, Junqueira, and Tall [1], Tall [57], [58], and [59]; Aurichi and Tall [8], Repovs and Zdomskyy [53], and Tall [61]. Alster and Hurewicz spaces are Menger, all productively Lindelöf spaces are Menger if and only if there exists a Michael space (in which case all productively Lindelöf Čech complete spaces are σ-compact - Tall [61]), and (as was mentioned earlier) a T3 Lindelöf space is Hurewicz if and only if in every Čech complete space Z in which it is contained there exists a σ-compact Y with X ⊆ Y ⊆ Z. Completeness is a generalization of compactness (compactness of X being completeness with respect to the completion structure whose only member is {X}) and thus complete spaces are candidates to replace compact spaces for use in distinguishing subspaces - complete Lindelöf spaces for our purposes. Given a subspace X of a completion space we describe X as completely distinguished by D if X is distinguished by D = {D : D ∈ D} and for each x ∈ X Dx = {D ∈ D : x ∈ D} is Cauchy and contains a complete member. Notice that if for some collection D the collection D distinguishes X and Φ is a collection of covers of X drawn from D, then each Dx is Cauchy, so D will distinguish X completely provided (enough) members of  D are complete with respect to the completion space ( D, Φ). 4.2 Theorem. If X is completely distinguished by a filtered family D, then D is focused at X. Proof. The proof is, in essence, the same as that of 3.5(a): For U an open set containing X, H the compliment  of U, x ∈ X, and some complete L ∈ Dx , the trace of Dx on H L is empty and must therefore include the empty set. (Otherwise as a Cauchy family of nonempty subsets of the complete space L the trace would have nonempty intersection.) Thus there exists Dx ∈ Dx with Dx ⊆ U, as desired.

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Theorem 4.2, in conjunction with 3.2 and 4.1, opens the doorway to distinguishing powerfully Lindelöf spaces using complete Lindelöf spaces in place of compact spaces. It is not clear if such an advance has any practical significance. As noted in 3.1(h), every Lindelöf T3 space X is distinguished by some family of Fσ (σ-compact) subspaces of βX, but Comfort has described, by simpler means, those spaces which can be identified using countable collections of members of the σ-lattice generated by the closed subsets of βX. The situation with respect to uncountable collections is no better: 4.3 Remark. If X is a normal Hausdorff space complete with respect to Φ, then for D the set of finite intersections of members of {P : P ∈ ϕ, ϕ ∈ Φ} with closure taken in βX, D (completely) distinguishes X. Thus among T3 spaces the class of Lindelöf spaces distinguished by families of closed compact subsets includes all complete spaces.   Proof. Frolik [24, Theorem 6] shows that for X complete normal Hausdorff, X = { ϕ : ϕ ∈ Φ}, so D distinguishes X. Since the members of D are compact Hausdorff, D distinguishes X completely. Use of non-compact spaces to distinguish powerfully Lindelöf spaces may have value among spaces which are not T3 , but the requirement in 3.3(a) that product sets be Wallace remains as a potential impediment, in essence restricting attention to P-spaces. The alternative condition in 3.3(a), that the filter bases in the distinguishing family be δ-stable, is related but less restrictive and is considered in the next section. 5. Countably stable filters Frolik [25] considers an approach different from “completion spaces”, replacing the layered Cauchy condition with a “depth” requirement that (using later terminology) the filter bases considered be “stable under countable intersections” (which we have abbreviated as δ-stable). Notice that a space is Lindelöf if and only if each δ-stable filter base of closed sets has nonempty intersection. Frolik’s focus is mainly on properties more general than Lindelöf (set in spaces more general than topological) but seem related to a class of spaces introduced in 1976 by Jerry Vaughan. A filter base is total if each finer filter base has nonempty adherence (intersection of the closures of its members), and of course a space is Lindelöf if each δ-stable filter base on it has nonempty adherence. Vaughan defines a space to be totally Lindelöf if each δ-stable filter base has a finer total δ-stable filter base. Total filters were first introduced by Pettis [51] and have also been called compactoid filters and compact filters and exploited in a wide range of applications by Mynard [40,41], and others. A filter base which converges to a compact set is total, and for regular spaces the converse is true Vaughan [67, Proposition 5.2]. Along with corresponding results for a wide range of compactness properties, Vaughan [67] showed that totally Lindelöf spaces are preserved by finite products; are productively and powerfully Lindelöf; include all σ-compact and Lindelöf P-spaces; and are σ-compact if and only if each compact subset is contained in a compact Gδ . Alster spaces, introduced in 1988, share these properties. A collection C of subsets of a space X is a k-cover if it covers not only the points of X, but also its compact subsets: K ⊆ X compact ∃C ∈ C with K ⊆ C. A space is Alster if each Gδ k-cover (that is, each k-cover made up of Gδ subsets) has a countable k-subcover. Alster [3] proved that these spaces are finitely productive, productively Lindelöf, and powerfully Lindelöf. Trivially, an Alster space with a k-cover of compact Gδ subsets is σ-compact. 5.1 Proposition. Each regular totally Lindelöf space is Alster. Proof. Suppose X is regular totally Lindelöf but has a Gδ k-cover G which has no countable subcover and set K = {X\G : G is the union of countably many members of G}. Note that K does not include the empty

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set and is therefore a δ-stable filter base. Since X is totally Lindelöf, K is contained in a finer total δ-stable filter base, and thus a total δ-stable filter I. Because X is regular, I converges to a compact set K - Vaughan [67, Proposition 5.2].  Since G is a Gδ k-cover there exists a set G = n Un in G with K ⊆ G and each Un open. Since I converges  to K there exist, for each n, Tn ∈ I with K ⊆ Tn ⊆ Un . Since I is δ-stable, T = n Tn is in I and hence, as a superset of T, G is in I. But X\G ∈ K ⊆ I, which contradicts the fact that I is a filter. Thus no such cover G exists, and therefore X is Alster. 5.2 Questions. (a) Is there an example of an Alster space which is not totally Lindelöf? (b) Is there an example of a totally Lindelöf (not regular) space which is not Alster? (c) What is the relationship of Alster/totally Lindelöf spaces to the spaces studied by Frolik in [25]? There is a group of product proofs which have the same general format: a property is characterized by guaranteed adherence for each of a class of “maximal” filters; it is shown that if F is such a filter on a product, then so too is each πα (F) on its factors; a point x = (xα ) is formed by choosing, for each α, xα in the adherence of πα (F), and maximality ensures (often by non-trivial argument) that x is an adherent point of F. The classic example of this format is Bourbaki’s proof of Tychonoff’s Theorem; Frolik’s proofs of “completeness” in Frolik [24] and proofs of countable productivity in Frolik [25] and Vaughan [67] are also of this type. Alexander’s proof of Tychonoff’s Theorem proceeds differently: Given a collection F with the finite intersection property of sets of the form πα−1 (Fβ ) where each Fβ is closed in the compact factor Xα , each    {Fβ } contains a point xα and (xα ) ∈ F so F is not empty. The basis for this proof is the fact that in proving compactness it suffices to consider such a family F, which Alexander proves in a more general setting by transfinite induction on an arbitrary well ordered collection with the finite intersection property. A different proof (Comfort and Negrepontis [18, Corollary 2.7], due to Tychonoff or perhaps Čech) uses  the Stone extension sα : βX → Xα of the projections πα : X = α Xα → Xα to create a map (s(x))α = sα (x) exhibiting X as a continuous image of a compact space. Such proofs can be used with other completeness properties, applying Fact 4.1(d) to a function with the required extension property. 6. Closed projections One could suggest that, with minor exception, “closed projections” are not particularly relevant to Lindelöf products, but I tend to view things from that perspective and in this case find such a view interesting, perhaps helpful. For X and Y Lindelöf XxY is Lindelöf if either of its projections is closed, and thus a Lindelöf space X is productively Lindelöf if each projection along X into Lindelöf Y is closed, i.e., if X is countably compact and therefore compact, and Lindelöf Y is productively Lindelöf if each projection along Lindelöf X into Y is closed, i.e. if Y is a Lindelöf P-space (see e.g., Noble [47, Corollary 2.3]). Thus one can view compact spaces and Lindelöf P-spaces as “corner points” in the range of productively Lindelöf spaces. Turning to infinite products, I will make use of a result which “bootstraps” closed projections on a product from those on subproducts: 6.1 Proposition. If for each α ∈ A the projection from   projection from β∈A Xβ to β<α Xβ .

 β≤α

Xβ to

 β<α

Xβ is closed, then so is each

The proof - at Noble [48, Theorem 1.8] - is by induction on the well ordered set of factors: given a point z in the closure of πF a point x in the closure of F is constructed such that π(x) = z. The induction step

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for non-limit ordinals uses the fact that if π : YxZ = Z is closed and z ∈ πF, there must exist a y such that (y, z) ∈ F. 6.2 Theorem. Let ℵ be an infinite cardinal, suppose R is a class of spaces which includes 1 and is closed under products of fewer than ℵ factors, and define P and Q as follows: • P = {X : π : XxY → Y is closed for each Y in R}; and  • Q = P R. If X =

 α∈A

Xα where each Xα is in Q, then

(a) if |A| < ℵ, X is in Q; and (b) if |A| = ℵ, X is in P. Proof. First note that the class Q is closed under finite products; indeed, for X ∈ P, and Y, Z ∈ R, πYxZ : XxYxZ → YxZ is closed, and for Y ∈ P, Z ∈ R, πZ : YxZ → Z and thus π : XxYxZ → Z is closed. Since this is true for each Z ∈ R. XxY is in P. Since X, Y ∈ Q implies also that XxY is in R, XxY is in Q. To complete the proof, note that X0 = X≤0 is in Q and suppose inductively that X<β is in Q for all β < α. If α = β + 1 for some such β, then X≤α is in Q as a finite product of members. Otherwise each π : X≤α → X<α is closed since Xα ∈ P and X<α ∈ R, so by Proposition 6.1, X is in P. If |α| < ℵ, X is also in R and thus Q. 6.3 Corollary. A countable product of Lindelöf P-spaces is Lindelöf. Proof. For R the class of P-spaces, which includes 1 and is closed under finite products, P is the class of Lindelöf spaces and Q is the class of Lindelöf P-spaces. A range of other examples is described by Vaughan [65, Theorem 2]. Readers interested in a categorical perspective could start with Clementino and Tholen [16]. 7. Small cylinders and hereditary properties Requiring that a projection be closed strongly limits generality. An alternative is to work with subsets which behave well with respect to projections, an obvious candidate being product sets. We explore that approach in this and the next section.   The span of a product set S = n Sn ⊆ n Xn is the largest n, n an integer if one exists, else ω, such that Sn = Xn . In geometric parlance, for n the finite span of a subset S the set B = π
 n

Xn is hereditarily Lindelöf if and only if each of its finite subproducts is hereditarily

Proof. Clearly if X is hereditarily Lindelöf its continuous images, and thus its factors, are hereditarily Lindelöf. Also, since a collection of open sets covers a subspace if and only if it covers a particular open subspace (its union), a space is hereditarily Lindelöf if and only if each of its open subspaces is Lindelöf. To

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  show that X = n Xn is hereditarily Lindelöf let U be an open subspace of X, U = n Un a span segregated cover of U by open product sets, and {Un } the associated segregation sequence. Since each Un is Lindelöf, each Un has a countable subcover, hence so does U. Essentially this same proof was used by Okuyama [50, Theorem 4.9] to show that a countable product is perfectly normal and paracompact provided each of its finite subproducts has those properties (a result credited to Morita). It was cited by Michael and used [43, Proposition 2.1(c)] to show that a countable product is Lindelöf provided each of its finite subproducts is Lindelöf and has each open subset an Fσ . (Which implies, of course, that they are hereditarily Lindelöf. Ernie was developing examples related to metrizable spaces where this open Fσ /closed Gδ condition is commonly satisfied.) The result for hereditarily Lindelöf spaces (as well as the corresponding result for hereditarily separable spaces, and several generalizations) is due to Zenor [70] by more or less this proof. In the same paper Zenor shows that C(X) is hereditarily Lindelöf in the pointwise topology whenever finite products of X are hereditarily separable (along with dual and much more general results) providing a wealth of examples more exotic than second countable spaces of well behaved hereditarily Lindelöf spaces. Michael’s example, [43, 1.3], of Xω and Yω hereditarily Lindelöf with Xω xYω not normal shows that powerfully hereditarily Lindelöf spaces need not, (CH), be productively Lindelöf. 8. Small cylinders and shrinkable products The key to the proof considered in Section 7 is that the members of a segregation sequence inherit the property of interest - in the case of Proposition 7.1, the Lindelöf property. Here we explore a condition insuring that segregation sequences can be refined by a suitable sequence of closed subspaces, permitting relaxation of “hereditary” to “closed hereditary”. A space is provided each countable open cover has an open refinement. The remark which follows is based upon Engelking [22, 5.2.1 and 5.2.3] and their proofs, which include separation assumptions not needed with our definitions. The results trace back to Dowker [19], Katětov [37], and Ishikawa [36]. 8.1 Remark. The following statements are equivalent: (a) X is . (b) Every countable open cover {Un } of X has a countable open refinement {Vn } with, for each n, Vn ⊆ Un . (c) Each countable increasing open cover {Un } has a countable refinement <{Fn }, {Vn }> with, for each n, . Proof. (a) (b). Given a countable open cover {Un } let V be a refinement and  for each V in V let m(V) be the smallest integer n such that V ⊆ Un . Then for Vn = {V : m(V) = n} {Vn } is an open refinement of {Un }. (b) (c). Given an increasing open cover {Un }, let {Wn } be an open refine ment with Wn ⊆ Un , set Fn = X\( {Wm } : m > n}, and set Vn = Int(Fn ). Then Fn ⊆ Um≤n Wn ⊆ Un so Fn and Vn are closed subsets of Un and since {Wn } is each point contained in some Fn . Thus <{Fn },{Vn }> is a cover and therefore a refinement of {Un }.  (c) (a). Given a countable open cover {Tn } let Un = i≤n Ti ; let <{Fn }, {Vn }> be refinement of {Un } with . Taking F−1 = ∅, set Wn = Un \ i
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 a countable open refinement of {Un }. Since Fn Wm = ∅ for m > n and each point is in some {Wn } is , as desired. A refinement such as {Vn } in (c) is called a shrinking of {Un } and the DeMorgan dual notion - for {Fn }  closed decreasing with empty intersection there exist open Gn with Fn ⊆ Gn and n Gn = ∅ - is called an expansion. A space both normal and countably metacompact is sometimes referred to as binormal and a space normal but not countably metacompact is called a Dowker space. Each regular Lindelöf space, being both normal and paracompact, is binormal, as is each normal countably compact space. 8.2 Observation. Any countable increasing open cover of X = composed of small open cylinders.

 n

Xn has a countable increasing refinement

Proof. Let {Un } be an increasing open cover; cover each Un with the open product sets which are contained  within it and let {Umn } be the resulting segregation series: Umn ⊆ Upn for m < p and m Umn = Un . Note that for n < q, since the starting open cover of Un is contained in that for Uq , one has Umn ⊆ Umq and  consequently Umn ⊆ Upp for p = max{m, n}. Thus n Unn = X so {Unn } is as desired.  8.3 Proposition. X = n Xn is countably paracompact if and only if each countable increasing open cover of X has a shrinking composed of small closed cylinders.  Proof. By 8.1, any increasing open cover {Un } of the countably paracompact space X = n Xn has a shrinking, an open cover {Vn } with Vn ⊆ Un . By the 8.2 {Vn } has a refinement {Wn } made up of small open cylinders. Then {Wn } is as desired since each Wn is a closed cylinder with Wn ⊆ Fn ⊆ Un . The converse follows from 8.1.  8.4 Theorem. X = n Xn is Lindelöf if each of its finite subproducts is Lindelöf and X is countably metacompact. For X regular, the converse is true.  Proof. Suppose X is countably metacompact, let U be an open cover segregated by span as n Un with segregation sequence {Un } and suppose {Fn } is a closed cover which refines {Un }. Since each Fn is Lindelöf each Un has a countable subcollection which covers Fn and thus U has a countable subcover. For the converse, if X is regular and Lindelöf it is normal and therefore countably metacompact. This proof, more or less, was introduced by Nagami [46] where it is used to prove substantially this result and product theorems for a number of other properties. (Nagami considered only Hausdorff spaces and posited countable paracompactness rather than countable metacompactness. Smirnov’s deleted sequence topology, Example 64 of Steen and Seebach Jr., [55], is a σ-compact Hausdorff space which is metacompact but neither countably paracompact nor regular.) What is required by this kind of proof for a property characterized in terms of small cylinder covers is that the property be closed hereditary, preserved by countable unions and refinements, and susceptible to the projection and extension manipulations involved. In particular, properties which transfer between {Un xX≥n } and {Un }. (Note that passing from an open cover to a cover of basic open subsets will usually destroy any cardinality restriction on the first cover, so by itself [a, b]-compactness is not a candidate when b = ∞.) Using a related proof, Zenor [69] showed, in essence (along with related results regarding perfect  normality), that a Hausdorff space X = n Xn is normal if and only if each finite subproduct is normal and X is countably paracompact.

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9. Supercovered spaces and finite Lindelöf products By “supercover” we refer to a collection C of subsets of a space X which not only cover X (each point in X is contained in a member of C) but cover a class S of subspaces of X: for each S ⊆ X with S ∈ S there exists a C ∈ C such that S ⊆ C. Such a collection is called an S-cover and for S a nonempty class of Lindelöf spaces which includes 1, a space X is (at least in some specific cases) called S-Lindelöf if each open S-cover of X has a countable S-subcover. The restrictions on S ensure that each open cover closed under the formation of countable intersections is an S-cover and each S-cover is a cover, so that S-Lindelöf spaces are Lindelöf. S-covers and S-Lindelöf spaces arise naturally in the study of function spaces with an S-open topology; for example, CS (X, Z) (the continuous functions from X to Z with the S-open topology) has countable tightness if and only if X is S-Lindelöf (some separation restrictions may apply - see for example Miller and Fremlin [44]). They also arise (e.g., Caserta, Di Maio, Kocinac, and Meccariello [15]) with the consideration of hyperspaces and certain selection principles relevant to game theory - and Lindelöf products. We will in particular be interested in n-covers (subspaces of cardinality less than or equal to the integer n), ω-covers (finite subspaces) and k-covers (compact subspaces - the covers already mentioned in the context of Alster spaces). Each second countable space is an ℵ0 -space (has a countable k-network) and each ℵ0 -space is k-Lindelöf. 9.1 Remark. Suppose S is a nonempty class of spaces closed under the formation of finite products and  continuous images. For X = i
Xn is Lindelöf if and only if X is n-Lindelöf; if X is n-Lindelöf and m < n, then X is m-Lindelöf; Xn is m-Lindelöf if and only if X is mn-Lindelöf; Xn is Lindelöf for each n if and only if X is ω-Lindelöf; if Xn is S-Wallace and S is closed under finite unions, Xn is S-Lindelöf if and only if X is.

Proof. (a). Notice that (x0 , . . . , xn−1 ) is in Vn if and only if {x0 , . . . , xn−1 } ⊆ V, so a collection V is an n-cover of X if and only if Vn = {Vn : V ∈ V} covers Xn . Suppose Xn is Lindelöf and let V be an open n-cover of X; then Vn is an open cover of Xn which, since Xn is Lindelöf, has a countable subcover {Wn : W ∈ W} for some countable W ⊆ V. Then W is an n-subcover of V, and therefore X is n-Lindelöf.

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Conversely, suppose X is n-Lindelöf and let U be an open cover of Xn which is closed under the formation of finite unions and set V = {open V ⊆ X : ∃U ∈ U with Vn ⊆ U}. It suffices to show that V is an n-cover of X, since then it will have a countable n-subcover W and Wn will be a countable cover which refines U, showing that U has a countable subcover and therefore that X is Lindelöf. To see that V is an n-cover, suppose S ⊆ X with |S| = n, and note that, since U is closed under finite  unions, ∃U ∈ U with Sn ⊆ U. By Wallace’s Theorem there exist open sets Vi such that Sn ⊆ i Vi ⊆ U and  hence for V = i Vi , Sn ⊆ Vn ⊆ U so S ⊆ V and therefore V∈ V. Thus V is an n-cover, as desired. (b) and (c) follow from (a) since subproducts of a Lindelöf product are Lindelöf, and (d) follows from (a)  since if for each n Un is a countable n-subcover of an ω-cover U of X, then n Un is a countable ω-subcover of x. (e). That Xn S-Lindelöf implies X S-Lindelöf was shown in 9.1. For the other direction, suppose X is S-Wallace and S-Lindelöf, U is an open S-cover of Xn , and V = {open V ⊆ X : Vn ⊆ U ∈ U}. If S ⊆ X with S ∈ S, then Sn ∈ S and therefore Sn ⊆ U for some U ∈ U. Since Xn is S-Wallace, there exist open Vi with   Sn ⊆ i Vi ⊆ U so for V = n Vn , Sn ⊆ Vn ⊆ U. Then S ⊆ V ∈ V, and therefore V is an S-cover of X. Since X is S-Lindelöf, V has a countable subcover W, and Wn is a cover of Xn . It remains to show that Wn  is an S-cover. If S ⊆ Xn is in S then {πi S} ⊆ S and, S being closed under finite unions, T = {πi S} is in S. Thus T ⊆ V for some V in the S-cover V and S ⊆ Tn ⊆ Vn which is in Wn , so Wn is an S-cover. The facts that Xn is Lindelöf for each integer n if and only if X is ω-Lindelöf and, implicitly, (a) through (c), are due to Gerlits and Nagy [26]. Earlier Arhangel’skii and Pytkeiev [7] equated “Xn Lindelöf for each n” with countable tightness of spaces of continuous functions on X in the pointwise topology. I have not found a reference for the case of k-covers in (e), but assume it is well known. 10. Supercovered spaces and countable Lindelöf products The importance of Alster spaces suggests interest in Gδ covers (covers composed of Gδ subsets). A space is said to be δ-Lindelöf if each Gδ cover has a countable subcover. For a space (X, τ ), δX = (X, δ(τ )) is the set X with the “Gδ topology” δ(τ ) generated by the Gδ subsets of (X, τ ). (Some authors use “Xδ ” or “bX” instead of δX; in Tychonoff spaces δ(τ ) coincides with the topology generated by the zero sets, which is called the Baire topology.) Lindelöf P-spaces are δ-Lindelöf and the coreflection X → δX of spaces to P-spaces carries δ-Lindelöf spaces to Lindelöf P-spaces. 10.1 Remark. Let G be the collection of Gδ subsets of a space (X, τ ). Then (a) (b) (c) (d) (e)

G is closed under the formation of countable intersections; thus G is a basis for δ(τ ); each Gδ in δX is open; thus δX is a P-space; a space X is δ-Lindelöf if and only if δX is Lindelöf; the δ-Lindelöf spaces coincide with the class of continuous images of Lindelöf P-spaces; and δ-Lindelöf spaces are finitely productive, productively Lindelöf, and powerfully Lindelöf.

   Proof. (a). If Gn = n Unm , then m Gn = n,m Unm .  (b). Since G is a basis, if x ∈ G = n Un with each Un ∈ δ(τ ) then for each n there exist Gn ∈ G with  x ∈ Gn ⊆ Un . By (a), V = n Gn ∈ G and x ∈ V ⊆ G, hence G is open. (c). A Gδ cover of (X, τ ) is a cover drawn from the basis G, i.e., is a basic open cover of δX. Every such cover has a countable subcover if and only if X is δ-Lindelöf/δX is Lindelöf. (d). A δ-Lindelöf X is the continuous image of the Lindelöf P-space δX. If f : X → Y is continuous with X   a Lindelöf P-space then Y is δ-Lindelöf: given a Gδ cover {Gα = n Uαn } of Y, { n f −1 (Uαn )} is an open cover of X which has a countable subcover {Hn } and {f(Hn )} is a countable subcover of {Gα }.

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(e). The class of Lindelöf P-spaces is finitely productive, productively Lindelöf and powerfully Lindelöf and therefore so is the class of their continuous images. That δX is a P-space is well known [39,35] as is, presumably, the rest of the remark. Note that δ-Lindelöf spaces (and thus a group of generalizations of scattered spaces which have been shown to be δ-Lindelöf: see Gewand [29], Hdeib and Pareek [33], Henriksen, Raphael, and Woods [35], and Barr, Kennison, and Raphael [12]) are contained in any class which includes the continuous images of Lindelöf P-spaces. For a different approach to spaces related to Lindelöf P-spaces see Hdeib [32]. Where S is a class of Lindelöf spaces which includes 1, call a space X Sδ -Lindelöf if each Gδ S-cover of X has a countable subcover. Note that Sδ -Lindelöf spaces are Lindelöf (because with members of S Lindelöf, each open cover closed under the formation of countable unions is a Gδ S-cover). For use in this context, call SxT δ-Wallace if G a Gδ with SxT ⊆ HxI ⊆ G implies there exist Gδ sets H and I with SxT ⊆ HxI ⊆ G. Note that Wallace subsets are δ-Wallace.  A proof by induction that a product α Xα has a property P provided each factor does involves (often, in part, implicitly) an induction step treating a product XxYxZ where X = X<α is assumed to possess P as part of the induction hypothesis, Y = Xα is hypothesized to have P, and Z = X>α is not yet known to enjoy P. Our next result, adapted from Alster’s proof that countable products of Alster spaces are Lindelöf, is such a step. It will be used to show, in particular cases, that a collection C with no countable subcover is itself not a cover. 10.2 Lemma. Let S be a class of spaces which includes 1, X and Y Sδ -Lindelöf spaces, and Z any space. If C is a collection of subsets of XxYxZ closed under the formation of countable unions which does not contain XxYxZ, then there exist subspaces S, T ∈ S with S ⊆ X and T ⊆ Y, such that if G and H are Gδ sets which contain S and T respectively, then GxHxZ is not contained in any member of C. Proof. First consider {S ⊆ X, S ∈ S : SxYxZ ⊆ C for some C ∈ C}. If each such S were contained in a Gδ set GS with GS xYxZ contained in some member of C, the sets {GS : S ⊆ X, S ∈ S} would be an S-cover of X (since 1 ∈ S they cover X) and would therefore have a countable subcover, forcing C to contain XxYxZ. Consequently there exists a set S ⊆ X, S ∈ S such that if G is a Gδ with S ⊆ G, then GxYxZ is not contained in any member of C. We repeat this argument with S and Y: consider {T ⊆ Y, T∈ S: SxTxZ ⊆ C for some C ∈ C}. If for each such T there exist Gδ sets GT and HT with S ⊆ GT , T ⊆ HT , and GT xHT xZ contained in some member of C, then {HT : T ⊆ Y, T ∈ S} is an S-cover of Y which has a  countable subcover {HT : T ∈ D} for some countable D ⊆ C. But then G = {GT : T ∈ D} is a Gδ and {GxHT xZ : T ⊆ Y, T ∈ S} is a countable cover of GxYxZ, forcing C to include a member containing GxYxZ. Since this contradicts the choice of S, there exists T ∈ S with T ⊆ Y and S and T as desired. 10.3 Theorem. Let L be the class of Sδ -Lindelöf spaces for S a class of Lindelöf spaces with 1 ∈ S. (a) If each product of n or fewer members of S is in S and is δ-Wallace, then each product of n or fewer members of L is in L. (b) if, for some space Z, SxZ is Lindelöf for each S ∈ S, then LxZ is Lindelöf for each L in L. (c) if each member of S is productively Lindelöf, then each member of L is productively Lindelöf. (d) if S is countably productive, each S-Wallace countable product of members of L is Lindelöf. Proof. (a). It suffices, by an induction halting at n (the case n = 1 being trivial) to consider two factors. Suppose X and Y are Sδ -Lindelöf, that G is a collection of Gδ subsets of XxY, and that the collection C, of unions of countably many members of G does not contain XxY. We show that G is not an S-cover. By Lemma 10.2 (with Z = 1 ignored) there exist S, T ∈ S with SxT ⊆ XxY such that if G ⊇ S and H ⊇ T are Gδ subsets, then GxH is not contained in any member of C. Since SxT is δ-Wallace, there is no Gδ set G

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with SxT ⊆ G∈ C. But G ⊆ C and since the induction has not halted, SxT is in S, so SxT is a member of S which is not contained in any member of G. Thus G is not an S-cover, so XxY is in L. (b). Suppose X is Sδ -Lindelöf, SxZ is Lindelöf for each S in S, and U is a collection of open subsets of XxZ closed under countable unions which does not contain XxZ. By 10.2 (with Y = 1 ignored) there exists an S ∈ S such that for each Gδ set G which contains S, no member of U contains GxZ, it follows that no member of U contains SxZ, and hence (since SxZ is Lindelöf and U contains countable unions) that U does not cover SxZ. Thus U is not a cover, and therefore XxZ is Lindelöf.  (c) follows from (b). (d). Consider an S-Wallace product L = n Xn and a collection U of open subsets of L closed under the formation of countable unions with L ∈ / U. Use of Lemma 10.2, both for the initial step (X = X0 , Y = 1, and Z = X>0 ) and for the induction step (X = Xn ), verifies  the existence of sets Sn ⊆ X with each Sn ∈ S such that if G = n Gn is a Gδ product set which contains  S = n Sn , then, for each n, no member of U contains G≤n xX>n . Since S ∈ S and L is S-Wallace (hence “δ-S-Wallace”), any open subset which contained S would contain such a set. Thus no member of U contains S, U is not a cover, and L is Lindelöf. 10.4 Examples. (a) (b) (c) (d)

S = {1}, Sδ -Lindelöf equals the class of δ-Lindelöf spaces; S = {finite subsets}, ω-Lindelöf ⊆ Sδ -Lindelöf ⊆ Alster spaces; S = {Lindelöf P-spaces}, Sδ -Lindelöf is productively Lindelöf; S = {compact spaces}, Sδ -Lindelöf = Alster spaces.

Note that if S ⊆ I then each I-cover is an S-cover so each Sδ -Lindelöf space is Iδ -Lindelöf; that is, larger classes S generate more general classes Sδ -Lindelöf. Where the questions have been decided, known productively Lindelöf spaces are both Alster and powerfully Lindelöf. Thus of the spaces in 10.3(c), the questions “Are they Alster?”, and “Are they powerfully Lindelöf?” are both interesting. 11. Expansion of powerfully Lindelöf classes In many cases a property which holds for a product holds for its factors, and this is usually the case for properties for which product theorems are framed (formulations such as “a product X has not P if and only if one of its factors has not P” being disfavored). Restricting attention to countable products, one might start to classify such formulations by complexity (more or less) as follows: (a) (b) (c) (d) (e) (f)

X has P if each factor has P; X has P if each factor has Q; X has P if each finite subproduct has P; X has P if each finite subproduct has Q; X has P if each finite subproduct has P and X has Q; X has P if each finite subproduct has Q and X has R.

We have seen examples of most of these. The product theorems for completion spaces discussed in Section 4 are of type (a) (as is that for second countable spaces). Corollary 6.3 (Lindelöf P-spaces) is type (b) and Proposition 7.1 (hereditarily Lindelöf spaces) fits type (c). Theorem 8.4 with, X metacompact, falls into type (e). This classification does not include a group of proofs based upon external characterizations of the factors, such as the proof of Tychonoff’s theorem mentioned in Section 3 which exhibits a product of compact spaces as the continuous image of a compact space. More generally, any process which preserves the Lindelöf

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property has the potential to “expand” a known class of spaces with Lindelöf products to a potentially larger such class, with the identification as an expansion serving as a proof of the related product theorem. My goal in this section is to begin an exploration of such proofs. Or rather to extend that exploration. Descriptive set theory, with its repeated countable unions and countable intersections, systematically develops such proofs, as does the Alexandroff-Arhangel’skii program investigating images and preimages of “nice” spaces under “nice” mappings. As applied to Lindelöf products, my goal is to create large powerfully Lindelöf classes by starting with simple classes and expanding them through the repeated use of “nice processes”. By a “process” I refer to any mechanism Ξ by means of which a given class S of spaces can be transformed to a second class Ξ(S). Examples include constructions (explicit or implicit) but also description by definition, for example, the “transformation” of the class of compact spaces to the class of k-spaces. I use Ξπ . Ξ , and Ξ∞ , to represent the formation of products of, respectively, finitely, countably, and arbitrarily many factors (taken from a given class, with repetitions allowed). Using similar notation, for S a class of spaces let • • • • •

Ξσ (S) Ξδ (S) ΞCS (S) ΞCI (S) ΞPP (S)

= {for each S in S, all countable unions of subspaces of S}; = {for each S in S, all countable intersections of subspaces of S}; = {all closed subspaces of members of S}; = {all continuous images of members of S}; and = {all perfect preimages of members of S}.

I add one more process to this collection of “usual suspects”. Let sS be the saturation of a subspace S and let ΞST (S) be the collection of “saturation twins” of subset members of S: • ΞST (S) = {Y : sY = sX, X, S ∈ S, X ⊆ S and Y ⊆ S}. Say that a process Ξ preserves a topological property P if Ξ(P) ⊆ P. Generalizing the term “powerfully Lindelöf”, for processes P and Q describe P as powerfully Q if each countable product of members of P is in Q, i.e., if Ξ (P) ⊆ Q. 11.1 Observation. Each of Ξσ , ΞCS , ΞCI , ΞST , and ΞPP preserve the Lindelöf property. Proof. With the exception of ΞST , these statements are well known, obvious, or both, the deepest being that products of perfect maps are perfect. That ΞST preserves the Lindelöf property is an immediate consequence of the fact that, as subspaces, saturation twins share exactly the same open covers. 11.2 Proposition. Let Ψ be one of {Ξπ , Ξ }, Ξ a composition of members of {ΞCS , ΞCI , ΞST , ΞPP }, and P a topological property which is preserved by Ξ. Then Ψ ◦ Ξ(P) ⊆ Ξ ◦ Ψ(P). Furthermore, if P is powerfully Q for some property Q then so is Ξ ◦ Ξ(P). Proof. If Ψ left commutes with A and B it left commutes with A ◦ B so it suffices to consider an individual  member Ξ of {ΞCS , ΞCI , ΞST , ΞPP }. For ΞCS , if each Xα is a closed subspaces of Yα ∈ P, then X = α Xα  is a closed subspace of α Yα , that is, Ψ ◦ ΞCS (P) ⊆ ΞCS ◦ Ψ(P). The same reasoning holds for ΞCI and ΞPP , since products of continuous functions are continuous and products of perfect maps are perfect. The same reasoning applies to ΞST because the saturation of a product is the product of the saturations  of its factors. To see this, let Xα be the saturation of Sα , (in some host we need not identify), X = α Xα ,  and S = α Sα . The product X is saturated, the intersection of {π −1 (Uα )} where each Uα is a collection of open subsets containing Sα with intersection Xα . Thus sS ⊆ X; to see that X ⊆ sS, suppose x ∈ X. Note that a point xα is in Xα = sSα if and only if there is a point sα in Sα such that every neighborhood

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of sα contains xα : if this were not the case, one could choose, for each s ∈ Sα , an open Us which did not contain xα and then form, as their union, an open set containing Sα but not xα , contradicting the fact that x is in sSα . Consequently we can choose s = (sα ) ∈ S so that every neighborhood of s contains x. In particular, any open set containing S contains x, so x ∈ sS, as desired. Changing “Ψ ◦ Ξ(P) ⊆ Ξ ◦ Ψ(P)” to “Ψ ◦ Ξ(P) ⊆ Ξ ◦ Ψ(Q)”, the same arguments show that if P is powerfully Q then so is Ξ ◦ Ξ(P). Where Ψ is a process (of interest here, Ξπ or Ξ ), Γ is a “generator” - a composition of finitely many other processes, and S is a “seed” class of spaces, define a class of spaces Ξ = Ξ[Ψ, Γ, S] inductively as:  Ξ0 = Ψ(S), Ξn+1 = Ψ ◦ Γ(Ξn ); and Ξ = n Ξn . When the values of Ψ, Γ, and/or S are established by the context they may (as was just done) be dropped from the notation, leading to terms such as Ξn [Ψ, S], Ξ[S], or just Ξ. Note that it is not required of the generator Γ that  ⊆ Γ() (as might reasonably be expected); some processes of interest do not satisfy that requirement. However, most eventually satisfy Ξn ⊆ Γ(Ξn ), or even terminate, so that eventually Ξn = Γ(Ξn ). While interested in more general instances, here I consider only the composition ΞPP ◦ ΞCS ◦ ΞCI ◦ ΞST . As an application of Proposition 11.2 we have: 11.3 Corollary. For Γ = ΞPP ◦ ΞCS ◦ ΞCI ◦ ΞST : (a) (b) (c) (d)

if if if if

Xn is Lindelöf for each n, then the same is true of members of Ξ[Ξπ , Γ, {X}]; P is a finitely productive class of Lindelöf spaces, then so is Ξ[Ξπ , Γ, P]; P is a countably productive class of Lindelöf spaces, then so is Ξ[Ξ , Γ, P]; and P is a powerfully Lindelöf class of spaces, then so is Ξ[Ξ , Γ, P].

Call a space X H-productively Lindelöf if XxH is Lindelöf for each hereditarily Lindelöf space H. Such spaces were studied by Alster [2] who showed that C-scattered Lindelöf T3 spaces are H-productively Lindelöf. (Introduced by Telgarsky [63], a space is C-scattered if each nonempty closed subset F contains a compact subset with its interior, taken relative to F, nonempty.) By analogy with one point compactifications of discrete spaces, let L(ℵ) be the space of cardinality ℵ with at most one non-isolated point p, it having neighborhoods with countable compliment. Notice that L(1) = 1, L(ℵ0 ) = N, and each L(ℵ) is C-scattered. The L(ℵ) generate interesting classes of spaces: 11.4 Examples of Ξ[Ψ, Γ, S] for Ψ = Ξ and Γ = ΞPP ◦ ΞCS ◦ ΞCI ◦ ΞST . (a) S = {1}: Ξ1 = K, the class of compact spaces, and expansion terminates because each of ΞPP, ΞCS , ΞCI , and ΞST , and hence Γ, preserve compactness. (b) S = {N}: Ξ0 includes N and Nω = P, the irrationals. Ξ1 adds closed subsets of Nω , σ-compact spaces  (and other perfect preimages), plus products Nx n Xn of members Xn . Frolik spaces are added with  Ξ2 . Since the map π : Nx n Xn → Σn Xn defined by π((n, x) = πn (x) ∈ Xn is continuous, Ξn will incorporate countable sums and their continuous images, countable unions. Also, since the diagonal of   n Xn is homeomorphic to n Xn , expansion will include countable intersections of subsets of Hausdorff members (whose diagonal is closed). Thus as expansion proceeds it will include the σ-lattice of subspaces of countable products, such as those considered by Comfort. (c) S = {L(ℵ)}: by Alster [2, Lemma 4] each scattered Lindelöf P-space of weight not greater than ℵ1 occurs as a closed subset of L(ℵ1 )ω and each space with a “co-countable point” - one each neighborhood of which has countable compliment - occurs as a continuous one to one image of an appropriate L(ℵ). (d) S = {Lindelöf T3 C-scattered spaces}: Alster [2, Remark 3] noted that the smallest class containing these spaces which is closed under the formation of closed subspaces, continuous images, and perfect preimages is H-productively Lindelöf and Barr, Kennison, and Raphael, [13, Theorem 25] show that Lindelöf T3 C-scattered (and more general) spaces are Alster.

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(e) S = {Alster spaces}: Alster spaces are preserved under the formation of closed subspaces, continuous images, and perfect preimages: Barr, Kennison, and Raphael, [13, Theorem 7] (proved there under a general assumption of Tychonoff separation which seems not to be used), and a space is Alster if and only if its saturation is Alster (see below). A countable product of Alster spaces need not be Alster, but by 10.2 the class of such products will be preserved by these processes and hence by Γ. Thus the expansion halts at Ξ1 . 11.5 Proposition. The saturation of an Alster space is Alster. Proof. It suffices to show that X and its saturation sX have the same Gδ k-covers, and since each Gδ k-cover of sX is such a cover of X, it suffices to prove the other direction. Let G be k-cover of X by Gδ subsets of the host and K a compact subset of sX. Define t : K → X by t(k) = k if k ∈ X and otherwise t(k) = x for some x ∈ X for which each neighborhood of x contains k (such an x must exist since k ∈ sX), and set T = t(K).  Then t: is continuous: if U is an open subset of K and x ∈ t−1 (U T), x is in each neighborhood of t(x)  so x is in U and therefore t−1 (U T) = U is open. Thus T is compact, hence contained in some member G of G. Since K is contained in any open set which contains T, it is contained in G, and thus G is a k-cover of sX, as desired. These examples only touch upon the possibilities: for any productively interesting class of Lindelöf spaces, identify which processes preserve it and use them, and minimal seed classes from which they can be generated, to expose the relationships among that and other such classes. There is no need to restrict attention to the processes considered here; indeed, many of the topics considered in earlier sections of this note or elsewhere in the literature could be viewed as or adapted to provide “processes” which could be explored in the context of expansion. This is, in particular, true of some techniques developed to construct counterexamples. Such processes offer the potential of identifying powerfully Lindelöf classes which include known “bad actors”. Of interest in the “bad actor” context is Okunev and Tamano [49] which examines generalizations of the Michael Line, producing many classes of spaces whose product with the space of irrationals is Lindelöf. Even more general possibilities are raised by Watson [68] which considers a class of counterexamples developed from or by analogy with the Alexandroff double arrow space. A program exploring techniques for proving countable products Lindelöf should include or be complimented by a corresponding consideration of techniques used to construct the related counterexamples. The references just mentioned represent significant contributions toward such a program, itself a component of the general consideration of counterexamples pioneered by Steen and Seebach Jr., [55]. References [1] Ofellia T. Alas, Leanardro F. Aurichi, Lucia R. Junqueira, Franklin D. Tall, Non-productive Lindelöf spaces and small cardinals, Houst. J. Math. 37 (2011) 1373–1381. [2] K. Alster, A class of spaces whose Cartesian product with every hereditarily Lindelöf space is Lindelöf, Fundam. Math. 114 (1981) 1173–1181. [3] K. Alster, On the class of all spaces of weight not greater than ω1 whose Cartesian product with every Lindelöf spaces is Lindelöf, Fundam. Math. 129 (1988) 133–140. [4] A.V. Arhangel’skii, On topological spaces which are complete in the sense of Čech, Vestn. Mosk. Univ. Ser. I Mat. Meh. (1961) 37–40 (Russian). [5] A.V. Arhangel’skii, Bicompact sets and the topology of spaces, Tr. Mosk. Mat. Obˆs. 13 (1965) 3–55, Russian, English translation: Trans. Mosc. Math. Soc. 13 (1965) 1–62. [6] A.V. Arhangel’skii, Paracompactness and metrization, in: A.V. Arhangel’skii (Ed.), The Method of Covers in the Classification of Spaces, in: General Topology, vol. III, Springer, 1995, pp. 3–70. [7] A.V. Arhangel’skii, E.G. Pytkeiev, Construction and clarification of topological spaces and cardinal invariants, Usp. Mat. Nauk 33 (1978) 29–84 (in Russian). [8] Leandro F. Aurichi, Franklin D. Tall, Lindelöf spaces which are indestructible, productive, or D, Topol. Appl. 159 (2011) 331–340.

N. Noble / Topology and its Applications 259 (2019) 287–310

309

[9] Leandro F. Aurichi, Lyubomyr Zdomskyy, Internal characterization of productively Lindelöf spaces, arXiv:1704.03843v2 [math.GN], 2017. [10] Richard N. Ball, Anthony W. Hager, Network character and tightness of the compact-open topology, Comment. Math. Univ. Carol. 947 (2006) 473–482. [11] Richard N. Ball, Anthony W. Hager, Charles W. Neville, The quasi-Fκ cover of compact Hausdorff space and the κ-ideal completion of an Archimedean l-group, in: R.M. Shortt (Ed.), General Topology and Its Applications, Proceedings of the 1988 Northeast Conference, in: Lecture Notes in Pure and Applied Mathematics, vol. 123, Marcel Dekker, New York, 1988, pp. 7–50. [12] Michael Barr, John F. Kennison, R. Raphael, Searching for absolute CR-epic spaces, Can. J. Math. (2007) 1–23. [13] Michael Barr, John F. Kennison, R. Raphael, On productively Lindelöf spaces, Sci. Math. Jpn. 65 (2007) 319–332. [14] Peter Burton, Franklin D. Tall, Productive Lindelöfness and a class of spaces considered by Z. Frolik, Topol. Appl. 159 (2012) 3097–3102. [15] A. Caserta, G. Di Maio, L.D.R. Kocinac, E. Meccariello, Applications of k-covers II, Topol. Appl. 153 (2006) 3277–3293. [16] M.M. Clementino, E. Giuli, W. Tholen, A functional approach to general topology, in: Categorical Foundations, in: Encyclopedia Math. Appl., vol. 97, Cambridge Univ. Press, Cambridge, 2004, pp. 103–163. [17] W.W. Comfort, Remembering Mel Henriksen and (some of) his theorems, Topol. Appl. 158 (2011) 1742–1748. [18] W.W. Comfort, S. Negrepontis, The Theory of Ultrafilters, Springer-Verlag, 1974. [19] C.H. Dowker, On countably paracompact spaces, Can. J. Math. 3 (1951) 219–224. [20] H. Duanmu, F.D. Tall, L. Zdomskyy, Productively Lindelöf and indestructibly Lindelöf spaces, Topol. Appl. 160 (2013) 2443–2453. [21] R. Engelking, On functions defined on Cartesian products, Fundam. Math. 59 (1966) 221–231. [22] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [23] Z. Frolik, On the topological product of paracompact spaces, Bull. Acad. Pol. Sci. Ser. Math. 8 (1960) 747–750. [24] Z. Frolik, On the descriptive theory of sets, Czechoslov. Math. J. 13 (88) (1963) 335–359, in English, Russian summary. [25] Z. Frolik, Prime filters with CIP, Comment. Math. Univ. Carol. 13 (1972) 553–575. [26] J. Gerlits, Z. Nagy, Some properties of C(X), I, Topol. Appl. 14 (1982) 151–161. [27] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislov, D.D. Scott, A Compendium of Continuous Lattices, SpringerVerlag, 1980. [28] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislov, D.D. Scott, Continuous Lattices and Domains, Cambridge University Press, 2003. [29] M. Gewand, The Lindelöf degree of scattered spaces and their products, J. Aust. Math. Soc. 37 (1984) 98–105. [30] Jean Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, Cambridge University Press, 2013. [31] Anthony W. Hager, Approximation of real continuous functions on Lindelöf Spaces, Proc. Am. Math. Soc. 22 (1969) 156–163. [32] H.Z. Hdeib, ω-Closed mappings, Rev. Colomb. Mat. XVI (1982) 65–78. [33] H.Z. Hdeib, C.M. Pareek, A generalization of scattered spaces, Topol. Proc. 14 (1989) 59–74. [34] M. Henriksen, J.R. Isbell, D.G. Johnson, Residue class Fields of lattice-ordered algebras, Fundam. Math. 50 (1961) 107–117. [35] M. Henriksen, R. Raphael, R.G. Woods, SP-scattered spaces; a new generalization of scattered spaces, Comment. Math. Univ. Carol. 48 (2007) 487–505. [36] F. Ishikawa, On countably paracompact spaces, Proc. Jpn. Acad. 31 (1955) 686–687. [37] M. Katětov, Measures in fully normal spaces, Fundam. Math. 38 (1951) 73–84. [38] K. Keimel, J. Paseka, A direct proof of the Hofmann-Mislove Theorem, Proc. Am. Math. Soc. 120 (1994) 301–303. [39] R. Levy, M.D. Rice, Normal P-spaces and Gδ topology, Colloq. Math. 47 (1981) 227–240. [40] F. Mynard, Products of compact filters and applications to classical product theorems, Topol. Appl. 154 (4) (2007) 953–968. [41] F. Mynard, Relations that preserve compact filters, arXiv:1002.3120v1 [math.GN], 2010. [42] Andrea Medini, Lyubomyr Zdomskyy, Productively Lindelöf spaces of countable tightness, arXiv:1405.6107v3 [math.GN], 2017. [43] Ernest A. Michael, Paracompactness and the Lindelöf property in finite and countable Cartesian products, Compos. Math. 23 (2) (1971) 199–214. [44] A.W. Miller, D.H. Fremlin, On some properties of Hurewicz, Menger, and Rothberger, Fundam. Math. 129 (1988) 17–33. [45] K. Nagami, Σ-spaces, Fundam. Math. 61 (1969) 169–192. [46] K. Nagami, Countable paracompactness of inverse limits and products, Fundam. Math. 73 (1972) 261–270. [47] N. Noble, Products with closed projections, Trans. Am. Math. Soc. 140 (1969) 181–191. [48] N. Noble, Products with closed projections II, Trans. Am. Math. Soc. 160 (1971) 169–183. This paper uses an invalid characterization of “n-m-compactness”, which property needs in several instances to be replaced with a sometimes stronger condition which Vaughan labels N[m,n]. See Stephenson and Vaughan [56] and Vaughan [64] for details. [49] Oleg Okunev, Kenichi Tamano, Lindelöf powers and products of function spaces, Proc. Am. Math. Soc. 124 (1996) 2905–2916. [50] A. Okuyama, Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 9 (1968) 236–254. [51] B.J. Pettis, Cluster sets of nets, Proc. Am. Math. Soc. 22 (1969) 386–391. [52] Paul Poncet, A class of compact subsets for non-sober topological spaces, arXiv:0912.5469v3 [math.GN], 2011. [53] D. Repovs, L. Zdomskyy, On th Menger covering property and D-spaces, Proc. Am. Math. Soc. 140 (2012) 1069–1074. [54] Marion Scheepers, Franklin D. Tall, Lindelöf indestructibility, topological games and selection principles, Fundam. Math. 210 (2010) 1–46. [55] Lynn A. Steen, J. Arthur Seebach Jr., Counterexamples in Topology, Holt, Rinehart and Winston, Inc., 1970. [56] R.M. Stephenson Jr., J.E. Vaughan, Products of initially m-compact spaces, Trans. Am. Math. Soc. 196 (1974) 177–189.

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[57] Franklin D. Tall, Productively Lindelöf spaces may all be D, arXiv:1104.2794v1 [math.GN], 2011, Can. Math. Bull. 56 (2013) 203–212. [58] Franklin D. Tall, Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D, Topol. Appl. 158 (2011) 2556–2563. [59] Franklin D. Tall, Some problems and techniques in set-theoretic topology, in: L. Balinkostova, A. Caicedo, S. Geschke, M. Scheepers (Eds.), Set Theory and its Applications, in: Contemp. Math., 2011, pp. 183–209. [60] Franklin D. Tall, Set theoretic problems concerning Lindelöf spaces, arXiv:1104.2796v1 [math.LO], 2011. [61] Franklin D. Tall, Definable versions of Menger’s conjecture, arXiv:1607.04781v1 [math.GN], 2016 (updated Jan 17, 2018). [62] Franklin D. Tall, Boaz Tsaban, On productively Lindelöf spaces, Topol. Appl. 158 (2011) 1239–1248. [63] R. Telgarsky, C-scattered and paracompact spaces, Fundam. Math. 73 (1971) 59–74. [64] J.E. Vaughan, Convergence, closed projections and compactness, Proc. Am. Math. Soc. 51 (1975) 469–476. [65] J.E. Vaughan, Some properties related to [a, b]-compactness, Fundam. Math. 87 (1975) 251–260. [66] J.E. Vaughan, Total nets and filters, in: Topology, Proc. Memphas State Univ. Conf., in: Lecture Notes, vol. 24, Marcel Dekker, 1976, pp. 259–265. [67] J.E. Vaughan, Products of topological spaces, Gen. Topol. Appl. 8 (1978) 207–217. [68] Stephen Watson, The construction of topological spaces: planks and resolutions, in: Miroslav Husek, Jan van Mill (Eds.), Recent Progress in General Topology, North Holland, 1992, pp. 672–757. [69] Phillip Zenor, Countable paracompactness in product spaces, Proc. Am. Math. Soc. 30 (1971) 199–201. [70] Phillip Zenor, Hereditary m-separability and the hereditary m-Lindelöf property in product spaces and function spaces, Fundam. Math. 106 (1980) 175–180.