A 0-dimensional, Lindelöf space that is not strongly D
aa r X i v : . [ m a t h . GN ] F e b A 0-DIMENSIONAL, LINDELÖF SPACE THAT IS NOTSTRONGLY D
DÁNIEL T. SOUKUP AND PAUL J. SZEPTYCKI
Abstract.
A topological space X is strongly D if for any neigh-bourhood assignment { U x : x ∈ X } , there is a D ⊆ X such that { U x : x ∈ D } covers X and D is locally finite in the topologygenerated by { U x : x ∈ X } . We prove that ♦ implies that there isan HFC w space in ω (hence 0-dimensional, Hausdorff and hered-itarily Lindelöf) which is not strongly D . We also show that any HFC space X is dually discrete and if additionally countable setshave Menger closure then X is a D -space. Introduction
A space X is said to be a D -space if for every neighbourhood as-signment { U x : x ∈ X } there is a closed discrete set D ⊆ X such that { U x : x ∈ D } covers the space [5] (we refer to D as a kernel for theneighbourhood assignment). One of the main open problems regardingtopological covering properties is whether every regular, Lindelöf spaceis a D -space. The latter question is due to E. van Douwen and we referthe reader to [7, 8, 9, 10] for more background.Recently, L. Aurichi [3] defined a space to be strongly D if for everyneighbourhood assignment { U x : x ∈ X } there is a set D ⊆ X suchthat { U x : x ∈ D } covers X and D is locally finite in the topologygenerated by { U x : x ∈ X } i.e., for each z ∈ X , there is a finite F ⊆ X such that z ∈ \ { U x : x ∈ F } and \ { U x : x ∈ F } ∩ D is finite.Note that if the topology generated by { U x : x ∈ X } is T (or if weadd the cofinite sets to this basis) then the above condition does implythat D is closed discrete in that topology. Aurichi has shown thatevery strongly D space is Lindelöf and that if there is a Lindelöf, T non strongly D -space then there is a Lindelöf, T non D -space. On the Date : February 19, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Lindelöf, covering property, D -space, HFC, dually dis-crete, ω -bounded, Menger, selection principle. other hand, in [16], we showed that under the assumption of ♦ , thereis a T , hereditarily Lindelöf non D-space. This provides the closestapproximation for a negative solution to van Douwen’s question todate.Now, we present a 0-dimensional, Hausdorff (and hence regular)modification of that example which is still hereditarily Lindelöf andnot strongly D . In fact, our space is homeomorphic to an
HFC w sub-space of ω which are basic examples of hereditarily Lindelöf spaces;Section 2 covers all the necessary definitions and the construction itself.We complement the previous result by showing that any HFC space(a natural strengthening of being
HFC w ) is dually discrete i.e., neigh-bourhood assignments always have discrete kernels (see Theorem 4).Moreover, we prove that any HFC space with the property that count-able sets have Menger closure is actually a D -space (see Theorem 7).The latter results are proved in Section 3. Figure 1 below summarizesthe known relations (where HL stands for hereditarily Lindelöf). HFCHFC w HLMenger strongly D D-space dually discrete
Theorem 4 × Theorem 1 + T ? Figure 1.
The implications between the covering propertiesThe red arrow marks van Douwen’s question and we conclude ourpaper with a list of further open problems.1.1.
Notation and terminology.
We let F n ( I, denote the set offinite partial function from I to 2. Any such finite function s determinesa basic open subset of the Tychonoff product I by setting [ s ] = { x ∈ I : x ⊃ s } .For a topological space ( X, τ ) , a local π -network at x ∈ X is a family F of arbitrary subsets of X so that for any open neighbourhood V of x , there is F ∈ F with F ⊂ V . A family of open sets U is an ω -cover of X if for any finite F ⊂ X , there is U ∈ U with F ⊂ U . Yet another version of this construction was used to show that the union of two D -spaces may not be D [17]. Acknowledgements.
D. T. Soukup would like to thank the gen-erous support of the Ontario Trillium Scholarship, FWF Grant I1921and NKFIH OTKA-113047 during the preparation of this manuscript.2.
The construction
Our goal is to prove the following.
Theorem 1.
Under ♦ , there is a 0-dimensional, Hausdorff and hered-itarily Lindelöf space which is not strongly D . We will define a topology ρ on ω by constructing U γ ⊂ ω for γ < ω .Sets of the form U s = \ γ ∈ s − (1) U γ ∩ \ γ ∈ s − (0) ( ω \ U γ ) will form the basis of the topology where s ∈ F n ( ω , . Our space ( ω , ρ ) naturally embeds into ω by the map β f ( β ) := x β ∈ ω defined by x β ( γ ) = ( if β ∈ U γ , if β / ∈ U γ where γ < ω . Condition (4) below in our inductive constructionensures that ρ is T and in turn, the map f is injective. For any s ∈ F n ( ω , , the intersection of the basic open set [ s ] in ω with f [ ω ] is exactly f [ U s ] . So f is a homeomorphism with its image.To ensure that ( ω , ρ ) is hereditarily Lindelöf, we employ the HFC w machinery from [11, Definition 3.2]. Let λ be some uncountable cardi-nal, F ∈ [ λ ] n and b ∈ n . We let F ∗ b denote the function from F to2 which which takes value b ( i ) on the i th element of F .Recall that X ⊂ λ is called an HFC w space if for any n < ω anduncountable, pairwise disjoint F ⊂ [ λ ] n , there is a countable F ⊂ F so that for any b ∈ n , | X \ [ { [ F ∗ b ] : F ∈ F }|≤ ω. Any
HFC w space is hereditarily Lindelöf [11, 3.3] and these spacesprovided some basic combinatorial examples of L -spaces (as one ofmany interesting applications). Let us refer the reader to I. Juhász’s[11] for more information on the structure and properties of such spaces.We will show in Proposition 2 that f maps ( ω , ρ ) to an HFC w spacein ω and so ρ is hereditarily Lindelöf. D. T. SOUKUP AND P. J. SZEPTYCKI
The sets U γ will be constructed simultaneously by an induction oflength ω . At each stage α ≥ γ , we will have an approximation U αγ for U γ and in fact, U γ ∩ ( α + 1) = U αγ . Moreover, we will assume γ + 1 ⊂ U γ so the subspace topology on α + 1 will be completely determined bystage α .Let us start the construction and assume ♦ . Let { B α : α ∈ ω } be a ♦ sequence capturing subsets of F n ( ω , i.e., B α ⊆ F n ( α, and for any B ⊆ F n ( ω , there are stationary many α such that B ∩ F n ( α,
2) = B α . Indeed, we will only be interested in capturinguncountable families B ⊆ F n ( ω , with pairwise disjoint domains inorder to assure that the space is HFC w . In addition, enumerate allcountable subsets of ω as { C α : α ∈ ω } so that for each α , we havethat sup C α < α . These sets are the potentially locally finite kernelsthat we should avoid.By recursion on α < ω , we define sets { U αγ : γ ≤ α < ω } so thatthe following inductive hypotheses are satisfied:(1) For all γ ≤ α , U αγ ⊆ α + 1 ,(2) For all α limit and n < ω , U α + nα + n = (0 , α ] ∪ { α + n } ,(3) For all γ ≤ α < β , U βγ ∩ ( α + 1) = U αγ ,(4) For all η < α there is γ < α such that η ∈ U αγ and α U αγ orvice-versa, η U αγ and α ∈ U αγ .We let τ α be the topology on α + 1 generated by the sets { U αγ : γ ≤ α } and let ρ α be the topology on α + 1 generated by the sets { U αγ , ( α + 1) \ U αγ : γ ≤ α } . Note that by (3), if α < β we have that τ α is the subspace topologyon α + 1 generated by τ β and similarly, ρ α is the subspace topology on α + 1 generated by ρ β . Moreover each ρ α is zero-dimensional and by(4) also T .To present the rest of the inductive hypotheses, we need some morenotation and definitions. For any finite s ∈ F n ( α, let U αs = \ ξ ∈ s − (1) U αξ ∩ \ ξ ∈ s − (0) (cid:0) α + 1 \ U αξ (cid:1) . These sets form a basis for ρ α . For all β ≤ α , let W αβ = { U αs : s ∈ B β } . We will make sure that if B β is large in some sense (see (6) below)then W αβ covers the interval ( β, α ] . This will help us prove hereditarilyLindelöfness through the HFC w machinery. Now, we have the following additional inductive hypotheses: for all β ≤ α ,(5) if C α is locally finite in τ α then S { U αξ : ξ ∈ C α } 6 = α + 1 ;(6) if B β ⊆ F n ( β, consists of functions with pairwise disjointdomains and if there is a countable elementary submodel M ≺ H ω that satisfies M ∩ ω = β and such that • { B γ : γ < ω } ∈ M , • there is an uncountable B ∈ M such that B β = B ∩ M ,and • there is ( V γ : γ < ω ) ∈ M such that V γ ∩ β = U αγ ∩ β forall γ < β ,then(a) if β ≤ α , then { dom ( s ) : s ∈ B β } is a local π -network at β in the τ α topology, and(b) if β < α then for each τ α -neighbourhood V of β and eachfinite subset F ⊆ V , the following is an ω -cover of ( β, α ] : { U αs : s ∈ B β and dom ( s ) ⊆ V \ F } . Let us carry out the construction and verify that the inductive hy-potheses can be preserved. First suppose that α < ω and for all η ≤ β < α , U βη has been defined satisfying the inductive hypotheses(1)-(6). Let U <αη = [ { U βη : η < β < α } and let τ ′ α denote the topology generated by these sets on α . Beforewe continue, note that for each β < α , τ β is the subspace topologyon β + 1 inherited from τ ′ α . Moreover, the key (6)(b) condition about ω -covers is satisfied by τ ′ α when restricted to the set ( β, α ) .Now, for each η < α , our goal is to extend U <αη to U αη by decid-ing whether to include or exclude α from it. Following our previousnotation, we let U <αs = \ ξ ∈ s − (1) U <αξ ∩ \ ξ ∈ s − (0) (cid:0) α \ U <αξ (cid:1) for s ∈ F n ( α, .First, let T be the set of β ≤ α satisfying the hypotheses of (6) andfirst assume that α ∈ T and T ∩ α = ∅ (hence α is a limit ordinal).Since α ∈ T , let M be the elementary submodel witnessing this andlet B ∈ M be the uncountable family with B ∩ M = B α .Next, fix an enumeration { ( β n , G n ) : n ∈ ω } of all pairs ( β, G ) where β ∈ T ∩ α and G is a finite subset of the interval ( β, α ) so that each such D. T. SOUKUP AND P. J. SZEPTYCKI pair appears infinitely often. For each β < α , we fix a decreasing localneighbourhood base { V n ( β ) : n < ω } in the τ ′ α topology. If C α is locallyfinite in α with respect to τ ′ α then we may assume that V ( β ) ∩ C α isfinite (for all β < α ).Now to proceed with the construction, we choose a sequence ( s n ) n ∈ ω by recursion on n such that(i) s n ∈ B β n and dom ( s n ) ⊆ V n ( β n ) ,(ii) dom ( s n ) ∩ C α = ∅ and (sup S k Proposition 2. For any uncountable family B ⊆ F n ( ω , with pair-wise disjoint domains, there is a countable B ′ ⊂ B so that ω \ [ { U F : F ∈ B ′ } is countable. Now the HFC w property is easily verified for the homeomorphic sub-space f [ ω ] of ω and so the topology ρ is hereditarily Lindelöf. Proof. Given B , we can find an elementary submodel M ≺ H (Θ) sothat M contains all the relevant parameters and for β = M ∩ ω , B ∩ F n ( β, 2) = B β . Indeed, there are club many models with all theparameters and B is guessed stationary often by the ♦ sequence. Inturn, at every stage α ≥ β , β satisfied the assumptions of condition (6)and so we made sure that S { U αs : s ∈ B β } covers ( β, α ] (see (6)(b)).Hence, ω \ [ { U s : s ∈ B β } ⊂ ω \ ( β + 1) , as desired. (cid:3) Proposition 3. The topology ρ is not strongly D .Proof. Indeed, this is witnessed by the neighbourhood assignment α U α . Suppose that C is locally finite in the topology generated by { U α : α < ω } . Since ρ is Lindelöf and C is locally finite in ρ too, C must becountable. Hence, there is an α < ω so that C = C α . In turn, at step α of the main induction, we made sure that α / ∈ U αξ = U ξ ∩ ( α + 1) for ξ ∈ C α (since C = C α was τ α locally finite at that point). So X = S ξ ∈ C α U α , as desired. (cid:3) This concludes the proof of Theorem 1. We remark that any count-able subspace of our topology ρ is second countable. In turn, our spaceis dually second countable i.e., any neighbourhood assignment has asecond countable kernel.3. HFC and D -spaces Our goal in this section is to analyse a strengthening of the HFC w property: a subspace X ⊆ λ is called HFC if for every n ∈ ω, b ∈ n and any infinite family F ⊂ [ λ ] n of pairwise disjoint sets, X \ [ { [ F ∗ b ] : F ∈ F } is countable [11, Definition 3.2]. Any HFC space is HFC w and sohereditarily Lindelöf as well. While HFC spaces are not necessarily left separated, in many caseswe get left separated spaces in the classical constructions of hereditarilyLindelöf spaces. It is well-known that every left-separated space is a D -space [9]. In general, we do not know whether all HFC spaces are D -spaces. However, we have the following result. Theorem 4. Any HFC space is dually discrete. Recall that X is dually discrete if for any neighbourhood assignment { U x : x ∈ X } there is a discrete D ⊆ X so that { U x : x ∈ D } covers X .It is also unknown whether all (hereditarily) Lindelöf space are duallydiscrete [1]. Proof. Given an HFC space X ⊂ λ , we can assume that neighbour-hood assignments are of the form N : X → F n ( λ, . We start bytaking a countable elementary submodel M ≺ H (Θ) (with Θ appro-priately large) so that X, N, λ ∈ M . Our first goal is to find a discrete D ⊂ M ∩ X so that X \ N [ D ] is countable.List all pairs ( ε, b ) ∈ F n ( λ ∩ M, × <ω as { ( ε ℓ , b ℓ ) : ℓ ∈ ω } suchthat each ( ε, b ) pair appears infinitely often. We construct a sequenceof points ( x n ) n<ω in M ∩ X so that(1) x n ∈ X ∩ M \ S { [ N ( x k )] : k < n } ,(2) M ∩ X ⊂ S { [ N ( x n )] : n ∈ ω } and(3) if n is even and there is x ∈ X \{ x k : k < n } so that(i) ε n ⊆ N ( x ) ,(ii) F = dom ( N ( x ) \ ε n ) is disjoint from dom ( N ( x k )) for k < n ,and(iii) N ( x ) \ ε n = F ∗ b n then we choose x n to be such.Note that condition (1) ensures that D = { x n : n < ω } is discrete.The construction is simple: at odd stages we work towards covering M ∩ X and at even stages, we see if condition (3) can be satisfies: ifso, we pick such an x n , otherwise an arbitrary one.We use N [ D ] to denote S { [ N ( x )] : x ∈ D } . Claim 5. | X \ N [ D ] |≤ ω .Proof. Suppose otherwise. Then there is Z ∈ [ X \ N [ D ]] ω and ( ε, b ) ∈ F n ( λ ∩ M ) × <ω such that for all x ∈ Z ,(1) N ( x ) ∩ M = ε , and(2) N ( x ) \ ε = dom ( N ( x ) \ ε ) ∗ b .Let Γ = { n ∈ ω : ( ε, b ) = ( ε n , b n ) } and recall the inductive construc-tion of the sequence { x n : n ∈ ω } . In particular, the set Z witnessesthat we were able to choose x n according to condition (3) when n ∈ Γ . In turn, there is an infinite set ˜ D ⊆ D so that ε ⊆ N ( x ) for each x ∈ ˜ D , { dom ( N ( x ) \ ε ) : x ∈ ˜ D } is pairwise disjoint and N ( x ) \ ε = dom ( N ( x ) \ ε ) ∗ b for all x ∈ ˜ D .Now, since X is an HFC space, X \ [ { [ dom ( N ( x ) \ ε ) ∗ b ] : x ∈ ˜ D } is countable. In particular, there is z ∈ Z such that z ∈ S { [ dom ( N ( x ) \ ε ) ∗ b ] : x ∈ ˜ D } . Pick x ∈ ˜ D such that z ∈ [ dom ( N ( x ) \ ε ) ∗ b ] . Recallthat z ∈ [ N ( z )] ⊂ [ ε ] and hence z ∈ [ ε ] ∩ [ dom ( N ( x ) \ ε ) ∗ b ] = [ N ( x )] .This contradicts z ∈ Z ⊆ X \ N [ D ] . (cid:3) Now, list X \ N [ D ] as { z n : n ∈ ω } (if N [ D ] already covers X thenthe proof is done). We define y n ∈ M ∩ X so that(3) z n ∈ [ N ( y n )] and(4) y n / ∈ S ℓ ≤ k n [ N ( x ℓ )] where k n is the maximum of n and min { k < ω : y n − ∈ N ( x k )] } . Whyis this possible? At step n , we consider the family of open sets { [ N ( y )] : y ∈ X \ [ ℓ ≤ k n [ N ( x ℓ )] } . Note that the latter is in M and since X is hereditarily Lindelöf, thereis a countable subfamily in M with the same cover. In turn, we canpick y = y n ∈ X ∩ M \ S ℓ ≤ k n [ N ( x ℓ )] which covers z n . Claim 6. { x n , y n : n < ω } is discrete.Proof. Simply note that S ℓ ≤ k n +1 [ N ( x ℓ )] is a neighbourhood of both x n and y n which contains only finitely many other x k , y k . (cid:3) This finishes the proof of the theorem since X = [ { [ N ( x n )] , [ N ( y n )] : n < ω } . ⊠ Our final theorem shows that in a class of ’locally small’ topologies,any HFC space must be a D -space. Theorem 7. Suppose that X ⊂ λ is HFC and the closure of everycountable subset of X is Menger. Then X is a D -space. The latter property is a natural variant of the well-studied ω -boundedness as-sumption i.e., countable sets having compact closure. See [12] for a fairly recentoverview. We thank L. Zdomskyy for recommending this version of our result; ouroriginal theorem assumed that countable sets in X have σ -compact closure. Recall that a space Y is Menger if for any countable sequence ofopen covers ( U n ) n ∈ ω , there are finite V n ⊆ U n such that S n ∈ ω V n covers Y . Let us refer the interested reader to [15] for background in selectionprinciples and topology.Any σ -compact space or Lindelöf space of size < d is Menger. Moreover, Aurichi showed that all Menger spaces are D -spaces [2,Corollary 2.7 ] and we need some of the topological games that heused in the proof. The partial open neighbourhood assignment game (or PONAG, for short) is played as follows. Our two players are N and C and N starts by playing a partial neighbourhood assignment { V x : x ∈ Y } for some Y ⊂ X covering X . Next, C replies by a D ⊂ Y closed discrete in X . In general, N plays { V x : x ∈ Y n } forsome Y n ⊂ X \ S { V x : x ∈ D k , k < n } so that { V x : x ∈ Y n } covers X \ S { V x : x ∈ D k , k < n } and C replies by D n ⊂ Y n closed discrete in X .Player N { V x : x ∈ Y } { V x : x ∈ Y } . . .Player C D ⊂ Y D ⊂ Y . . .Player C wins if S { V x : x ∈ D n , n < ω } covers X .We will use the fact that if Y is Menger then N has no winningstrategy in PONAG [2, Proposition 2.6]; from this, Y being a D -spaceeasily follows. In fact, Aurichi’s proves in [2, Proposition 2.6] that N has no winning strategy in the following modification PONAG fin ofthe original PONAG game: N plays as before but C is only allowedto reply by finite sets (instead of arbitrary closed discrete ones). Thewinning condition is the same as before. This minor modification isquite important in our following proof (and also shows that Mengerspaces are strongly D ). Proof of Theorem 7. First, any neighbourhood assignment (after someshrinking) can be coded by a map N : X → F n ( λ, so that x ∈ [ N ( x )] .We use N [ E ] to denote the set S { [ N ( x )] : x ∈ E } in short.Our plan is to find a closed discrete set D such that X \ N [ D ] iscountable. Since countable spaces are D -spaces, we can find a closeddiscrete ˜ D ⊂ X \ N [ D ] so that N [ D ∪ ˜ D ] = X , as desired (note thatthe union D ∪ ˜ D is still closed discrete).Now, let M ≺ H (Θ) be a countable elementary submodel (with Θ appropriately large) so that X, N, λ ∈ M . Let Y = X ∩ M and notethat Y is Menger.We construct a sequence ( ˆ D n ) n<ω of finite sets in M ∩ X so that Here d denotes the dominating number [4]. (1) Y ⊂ N [ ˆ D ] for ˆ D = S { ˆ D n : n ∈ ω } , and(2) ˆ D n ∩ N [ ˆ D n − ] = ∅ .We will find these sets using the fact that player N has no winningstrategy in the PONAG fin game on Y . So, we define a strategy σ for N as follows. We set V x = [ N ( x )] and let N start by playinga cover ( V x ) x ∈ Y of X for some countable Y ∈ M . This is possiblesince X is Lindelöf (and we can choose the witness for that in M byelementarity) and so Y ⊂ M as well. In general, given a partial gameplay Y , ˆ D , Y , ˆ D , . . . , Y n − , ˆ D n − so that Y k ∈ M , σ will simply play ( V x ) x ∈ Y n where Y n ⊂ X \ S k PONAG fin game so that the sets ˆ D n are in M too. In fact, the open sets corresponding to the extra points x n we chosecover X modulo a countable set. The proof of this claim is exactly asthe proof of Claim 5 which we omit repeating.This concludes the proof of the theorem. ⊠ We do not know how much the assumption of X being HFC can beweakened or if the local smallness assumption can be dropped.4. Open problems The main problem of van Douwen remains open. Problem 10 (van Douwen) . Is there a regular, Lindelöf but non D -space? Surprisingly, the following question seems to be open as well (see[1]). Problem 11. Is there a T , hereditarily Lindelöf but non dually dis-crete space? We should emphasise that even consistent examples would be verywelcome.Regarding Theorem 7 and Theorem 4, we ask the following. Problem 12. Is every HFC space a D -space? Problem 13. Is every HFC w space dually discrete? Problem 14. Suppose that X is an HFC space and every countablesubset of X has Menger closure. Is X strongly D ? Problem 15. Suppose that every countable subset of X has Mengerclosure. Is X a D -space whenever it is(a) (hereditarily) Lindelöf, or(b) HFC w ? It would be equally interesting to see an answer for the precedingquestions if instead of the Menger property we assume that countablesets have σ -compact closure. Finally, let us refer the reader to [9] formany more interesting open problems around D -spaces. So in the end, the Menger machinery is used to make this sequence closeddiscrete and the HFC property to cover X . References [1] Alas, Ofelia T., Lucia R. Junqueira, and Richard G. Wilson. Dually discretespaces Topology and its Applications 155.13 (2008): 1420-1425.[2] Aurichi, Leandro F. D-spaces, topological games, and selection principles Topology Proc. Vol. 36. 2010.[3] L.F. Aurichi, D-spaces, separation axioms and covering properties [4] Blass, Andreas. Combinatorial cardinal characteristics of the continuum Hand-book of set theory. Springer, Dordrecht, 2010. 395-489.Houston Journal of Mathematics, No. 3 (2011) 1035-1042.[5] E.K. van Douwen and W.F. Pfeffer, Some properties of the Sorgenfrey line andrelated spaces Pacific Journal of Mathematics, No.2 (1979) 371-377.[6] A. Dow An introduction to applications of elementary submodels to topology Topology Proc. 13 (1988), no. 1, 17–72.[7] T. Eisworth On D-spaces in Open Problems in Topology, Elsevier, (2007) 129-134.[8] W.G. Fleissner and A.M. Stanley D-spaces Topology and its Applications 114(2001) 261–271.[9] G. Gruenhage A survey of D-spaces , Contemporary Mathematics, to appear.[10] M. Hrušák and J.T. Moore Introduction: Twenty problems in set-theoretictopology in Open Problems in Topology, Elsevier, (2007) 111-113.[11] Juhász, István. HFD and HFC type spaces, with applications Topology and itsApplications 126.1-2 (2002): 217-262.[12] Juhász, István, Jan van Mill, and William Weiss. Variations on ω -boundedness Israel Journal of Mathematics 194.2 (2013): 745-766.[13] K. Kunen Set Theory, An Introduction to Independence Proofs Studies in Logicand the Foundations of Mathematics v102, North-Holland, 1983.[14] J. van Mill, V.V. Tkachuk and R.G. Wilson Classes defined by stars and neigh-bourhood assignments Topology Appl. 154 (2007), 2127–2134.[15] Scheepers, Marion. Selection principles and covering properties in topology Note di Matematica 22.2 (2003): 3-41.[16] Soukup, Dániel T., and Paul J. Szeptycki. A counterexample in the theory ofD-spaces Topology and its Applications 159.10-11 (2012): 2669–2678.[17] Soukup, Dániel T., and Paul J. Szeptycki. The union of two D-spaces need notbe D Fundamenta Mathematicae 220.2 (2013): 129-137.(D. T. Soukup) Kurt Gödel Research Center for Mathematical Logic,Faculty of Mathematics, University of Vienna, WÃďhringer Strasse25, 1090 Wien, Austria E-mail address , Corresponding author: [email protected] URL : ∼ soukupd73/ (P. J. Szeptycki) Department of Mathematics and Statistics, Facultyof Science and Engineering, York University, Toronto, Ontario, CanadaM3J 1P3 E-mail address ::