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On the D-property of certain products
Topology and its Applications 195 (2015) 297–311
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Topology and its Applications www.elsevier.com/locate/topol
On the D-property of certain products ✩ Yasushi Hirata, Yukinobu Yajima ∗ Department of Mathematics, Kanagawa University, Yokohama, 221-8686 Japan
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Article history: Received 18 May 2014 Received in revised form 2 September 2014 Available online 2 October 2015 MSC: primary 54B10, 54D20 secondary 03E10, 03E75
a b s t r a c t We first point out that the product of a subparacompact DC-like space and a subparacompact D-space is a D-space. As a main result, we use elementary submodels to prove that any countable product of subparacompact DC-like spaces is a D-space. More generally, we prove that any uncountable product of subparacompact DC-like spaces is a D-space if and only if all but countably many of the factors are compact. © 2015 Elsevier B.V. All rights reserved.
Keywords: D-space DC-like Subparacompact Submetacompact Elementary submodel
1. Introduction An open neighborhood assignment for a space X is a function ϕ : X → τ (X) such that x ∈ ϕ(x) for each x ∈ X, where τ (X) denotes the topology of X. A space X is called a D-space (or has the D-property) [5] if for each open neighborhood assignment ϕ for X, there is a closed discrete subset D in X with {ϕ(x) : x ∈ D} = X. The concept of D-spaces was introduced by van Douwen and Pfeffer [5] and they proved that every finite power of the Sorgenfrey line S is a D-space. Since then there have been very many papers devoted to the study of D-spaces. In particular, there is a survey by Gruenhage [8] which gives a nice overview of D-spaces. The D-property is a covering property. It is natural to consider whether given covering properties are productive. However, the D-property is not even finitely productive. In fact, it was shown in [1] that there are a Lindelöf D-space X and a separable metric space M such that X × M is not a D-space. On the other hand, Peng [13] proved that any countable product of paracompact DC-like spaces is a D-space. ✩
This research was supported by Grant-in-Aid for Scientific Research (C) 24540147.
* Corresponding author. E-mail addresses: [email protected] (Y. Hirata), [email protected] (Y. Yajima). http://dx.doi.org/10.1016/j.topol.2015.09.036 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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First, we point out that the product of a subparacompact DC-like space and a subparacompact D-space is a D-space. Second, we show that the product of a hereditarily submetacompact C-scattered space and an arbitrary D-space is a D-space. Third, we also show that the product of a paracompact subspace of an ordinal and a submetacompact D-space is a D-space The first result gives us motivation to extend Peng’s result above by replacing paracompactness with subparacompactness. Here it is proved by using an elementary submodel that every countable product of subparacompact DC-like spaces is a D-space. It has been suggested in [17] that elementary submodels will be very useful to study covering properties. This result may be considered as an illustration of this. Moreover, for the uncountable product case, we prove that the uncountable power of an infinite countable discrete space is not a D-space. Finally, we raise several unsolved problems. All spaces here are assumed to be regular T1 . We denote by ω the set of all natural numbers. 2. Topological games Let K be a non-void class of spaces. For such a class K, we denote by DK the class of all spaces which have a discrete (clopen) cover by members of K. For example, when 1 denotes the class of all one-point spaces, D1 means the class of all discrete spaces. As such a K, we mainly use the two classes C and D as follows: • C denotes the class of all compact spaces and • D denotes the class of all D-spaces. In 1975, Telgársky introduced and studied the topological game G(K, X). Definition 1. ([20]) Let X be a space. Let K be a class of spaces which is hereditary with respect closed sets (i.e., for each closed set F in Y ∈ K, F ∈ K holds). In the game G(K, X), two players choose closed subsets En and Fn in X for each n ∈ ω \ {0} in turn E1 , F1 , E2 , F2 , · · ·. Player I chooses En ∈ K with En ⊂ Fn−1 , where F0 = X. Player II chooses Fn with Fn ⊂ Fn−1 \ En . Player I wins if n∈ω Fn = ∅. Player II wins otherwise. A space X is said to be K-like if Player I has a winning strategy in the game G(K, X). Theorem 2.1. (Galvin and Telgársky [7]) A space X is K-like if and only if there is a function s : 2X → 2X ∩ K, where 2X denotes the family of all closed sets in X, such that (i) s(F ) ⊂ F for each closed set F in X, (ii) if {Fn }n∈ω is a decreasing sequence of closed sets in X satisfying that s(Fn−1 ) ∩ Fn = ∅ for each n ∈ ω, where F0 = X, then n∈ω Fn = ∅ holds. Such a function s above is called a stationary winning strategy for Player I in G(K, X). We can arbitrarily take the first choice of Player I for the s as follows. Proposition 2.2. Let X be a K-like space with a non-empty closed set E in X with E ∈ K. Then there is a stationary winning strategy s for Player I in G(K, X) such that (iii) s(X) = E, (iv) s(F ) is non-empty for each F ∈ 2X \ {∅}. Proof. By Theorem 2.1, we can take a stationary winning strategy s0 for Player I in G(K, X). We define a function s : 2X → 2X ∩ K as follows; for each F ∈ 2X meeting E, let s(F ) = E ∩ F , and for each F ∈ 2X
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disjoint from E, let s(F ) = s0 (F ) if F ∩ s0 (X) = ∅ and let s(F ) = F ∩ s0 (X) if F ∩ s0 (X) = ∅. Then s satisfies (iii) and (iv). Let s(X), F1 , s(F1 ), F2 , s(F2 ), · · · be a play in G(K, X). When F1 ∩ s0 (X) = ∅, s0 (X), F2 , s0 (F2 ), F3 , s0 (F3 ), · · · is a play in G(K, X). Hence we have n∈ω Fn = n≥2 Fn = ∅. When F1 ∩ s0 (X) = ∅, s0 (X), F1 , s0 (F1 ), F2 , s0 (F2 ), · · · is a play in G(K, X). Hence we have n∈ω Fn = ∅. Thus s is a stationary winning strategy for Player I in G(K, X). 2 3. Subparacompact DC-like spaces Let X × Y be a product of two spaces. A subset of the form E × F in X × Y is called a rectangle. A rectangle E × F is called a closed rectangle in X × Y if E and F are closed in X and Y , respectively. Definition 2. ([21]) A product X × Y of two spaces is called a D-product if for any disjoint pair E, F of closed sets E and F in X × Y , there is a σ-discrete collection F by closed rectangles in X × Y such that F ⊂ F ⊂ (X × Y ) \ E. For two classes K1 and K2 of spaces which are hereditary with respect closed sets, we put K1 × K2 = {E : E is a closed set in X × Y with X ∈ K1 and Y ∈ K2 }. The following result was proved in [21, Theorem 2.1] (see [23, Theorem 5.3] for its improved proof). Theorem 3.1. ([21]) Let X and Y be a K1 -like space and a K2 -like space, respectively. If X ×Y is a D-product, then it is D(K1 × K2 )-like. Recall that a space X is subparacompact if every open cover of X has a σ-locally finite (or σ-discrete) closed refinement. A typical example of D-products was given as follows. Proposition 3.2. ([22]) If X is a subparacompact DC-like space and Y is a subparacompact space, then X × Y is a subparacompact D-product. The following is due to Peng, which was also stated in [8, Theorem 3.3] with a proof. Lemma 3.3. ([12]) Every D-like space is a D-space. By DC ⊂ D, note that every DC-like space is a D-space. Lemma 3.4. ([2]) If K is a compact space and Y is a D-space, then K × Y is a D-space. Lemma 3.5. Let X be a DC-like space and Y a D-space. If X × Y is a D-product, then it is a D-space. Proof. Considering K1 = DC and K2 = D, it follows from Theorem 3.1 that X × Y is D(DC × D)-like. By Lemma 3.4, we have C × D ⊂ D. So we should notice that D(DC × D) = D(D(C × D)) ⊂ D(DD) = DD = D. Hence X × Y is D-like. By Lemma 3.3, X × Y is a D-space.
2
The following is an immediate consequence of Proposition 3.2 and Lemma 3.5. Corollary 3.6. If X is a subparacompact DC-like space and Y is a subparacompact D-space, then X × Y is a D-space.
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4. Hereditarily submetacompact C-scattered spaces Recall that a space X is submetacompact if for every open cover U of X, there is a sequence {Vn }n∈ω of open refinements of U, satisfying that for each x ∈ X, one can find nx ∈ ω such that Vnx is point-finite at x. Moreover, recall that a cover A of a space X is directed if for any A0 , A1 ∈ A, there is A2 ∈ A with A0 ∪ A1 ⊂ A2 . Using this, Junnila proved the following useful characterization of submetacompactness. Theorem 4.1. ([10, Theorem 4.4]) A space X is submetacompact if and only if every directed open cover of X has a σ-closure-preserving closed refinement. Recall that a space X is C-scattered if every non-empty closed subset F in X has a point with a compact neighborhood in F . For each F ∈ 2X \ {∅}, let F ∗ = {x ∈ F : x has no compact neighborhood in F }. Let X (0) = X and X (α+1) = (X (α) )∗ for each ordinal α. For a limit ordinal λ, let X (λ) =
α<λ
X (α) .
Fact 4.2. ([19]) Let X be a C-scattered space. Then the following are true. (i) X (α) = ∅ for some ordinal α, (ii) X (α) is locally compact if X (α+1) = ∅, (iii) each open (or closed) set in X is C-scattered. It follows from [20, Theorem 9.7] that a subparacompact C-scattered space is DC-like. It is easily seen by Theorem 3.1 and Proposition 3.2 that the subparacompact DC-like property is finitely productive. So Corollary 3.6 is a generalization of [11, Theorem 2.1] and [14, Corollaries 20 and 21]. The following seems to be known. However, for making sure, we put a simple proof. Lemma 4.3 (Folklore). If a space X is locally compact and submetacompact, then it has a σ-closure-preserving cover by compact sets. Proof. Let U be the family of all open sets U in X such that U is compact, then it is a directed open cover of X. By applying Theorem 4.1, we obtain a σ-closure preserving closed refinement F of U. Obviously, F is a cover of X by compact sets. 2 We also use the following, which was generalized in [9, Proposition 5.7]. Lemma 4.4. (Peng [12]) If a space X has a σ-closure-preserving closed cover by D-subspaces, then it is a D-space. Recall that a space X is hereditarily submetacompact if each subspace of X is submetacompact. Proposition 4.5. If X is a hereditarily submetacompact, C-scattered space and Y is a D-space, then X × Y is a D-space. Proof. By Fact 4.2(i), X (α) = ∅ for some ordinal α. We show by induction with respect to α. When α = 0, this is obvious because X = X (0) = ∅. Take an ordinal α, and assume that the proposition is true if X (β) = ∅ for β < α.
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Case 1. Let α be a successor. Let α = δ + 1. Since X (δ) is locally compact and submetacompact by Fact 4.2(ii), it follows from Lemma 4.3 that X (δ) has a σ-closure-preserving cover K by compact sets. Then {K × Y : K ∈ K} is a σ-closure-preserving closed cover of X (δ) × Y by D-subspaces. By Lemma 4.4, X (δ) × Y is a D-space. Take any closed subset E in X × Y disjoint from X (δ) × Y . By Fact 4.2(iii), X \ X (δ) is hereditarily submetacompact and C-scattered with (X \ X (δ) )(δ) = ∅. So it follows from the inductive assumption that (X \ X (δ) ) × Y is a D-space. Since E is a closed subset in (X \ X (δ) ) × Y , E is also a D-space. Hence X × Y is D-like. By Lemma 3.3, X × Y is a D-space. Case 2. Let α be a limit ordinal. Let U = {X \ X (β) : β < α}. Since U is directed, it follows from Theorem 4.1 that U has a σ-closure-preserving closed refinement F. Then note that each F ∈ F is hereditarily submetacompact and C-scattered such that F (βF ) = ∅ for some βF < α. By the inductive assumption, each F × Y is a D-space. Since {F × Y : F ∈ F} is a σ-closure-preserving closed cover of X × Y by D-subspaces, it follows from Lemma 4.4 that X × Y is a D-space. 2 5. Paracompact subspaces of an ordinal The following seems to be known. However, we state its proof for the reader’s convenience. Lemma 5.1 (Folklore). Let A be a subspace of an ordinal. Then the following are equivalent. (a) A is paracompact. (b) A is DC-like. (c) There is not an ordinal μ ∈ / A with cf μ > ω such that A ∩ μ is stationary in μ. / A with cf μ > ω such that A ∩ μ is stationary in μ. Since A ∩ μ Proof. (a) → (c): Assume that there is a μ ∈ is clopen in A, it is paracompact. Let U = {A ∩ [0, ξ] : ξ ∈ A ∩ μ}. Then U is an open cover of A ∩ μ. By the standard way using the Pressing Down Lemma, we see that U has no point-countable open refinement. This is a contradiction. (c) → (b): Assume that A is not DC-like. Take the smallest μ such that A ∩ [0, μ] is not DC-like. It is easy to see that μ ∈ / A. By the assumption, cf(μ) ≤ ω or A ∩ μ is not stationary in μ. In any case, we can represent A ∩ μ as ν∈cf(μ) Bν such that Bν ⊂ A ∩ [0, ξν ] for some ξν < μ. By the minimality of μ, each Bν is DC-like. Hence A ∩ μ = A ∩ [0, μ] is DC-like. This is a contradiction. (b) → (a): Since A is collectionwise normal, this immediately follows from [20, Theorem 10.6]. 2 Using the results above, we show the following. Proposition 5.2. If A is a paracompact subspace of an ordinal and Y is a submetacompact D-space, then A × Y is a D-space. Proof. Let A be a paracompact subspace of an ordinal μ ˆ, and Y a submetacompact D-space. For each μ≤μ ˆ, put Aμ = A ∩ [0, μ]. And put A−1 = ∅. Obviously, A−1 × Y = ∅ is a D-space and Aμˆ = A holds. By induction on an ordinal μ ≤ μ ˆ, we will prove that Aμ × Y is a D-space. After finishing induction, we conclude that A × Y = Aμˆ × Y is a D-space. Let μ be an ordinal and assume that Aν × Y is a D-space for each ν < μ. We show that Aμ × Y is a D-space. Case 1. μ ∈ / A. Since A is paracompact, we have that cf μ ≤ ω or Aμ = A ∩ μ is non-stationary in μ with cf μ > ω. Hence Aμ = j∈J Bj for some collection {Bj : j ∈ J} of subspaces of Aμ such that for each j ∈ J, there is
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a νj < μ with Bj ⊂ Aνj . Then Aνj × Y is a D-space by the inductive hypothesis, so Bj × Y is a D-space since it is a closed subspace of Aνj × Y , hence Aμ × Y = j∈J (Bj × Y ) is a D-space. Case 2. μ ∈ A. Let ϕ be an open neighborhood assignment for Aμ × Y . It suffices to find a closed discrete subset D in Aμ × Y such that Aμ × Y = {ϕ(d) : d ∈ D}. Put Z0 = {μ} × Y . Then Z0 is a closed set of Aμ × Y which is homeomorphic to Y . Since Y is a D-space, we can take a closed discrete subset D0 in Z0 (also in Aμ × Y ) such that D0 ⊂ Z0 ⊂ W0 , where let W0 = {ϕ(d) : d ∈ D0 }. Now, let Q = {Q ⊂ Y : Q is open in Y such that V × Q ⊂ W0 for some open neighborhood V of μ in Aμ }. Since Q is a directed open cover of Y which is submetacompact, it follows from Theorem 4.1 that Q has a σ-closure preserving closed refinement F. Enumerating members of F in natural, we have F = {Fξ : ξ < λ} for some ordinal λ and F<ξ := ζ<ξ Fζ is closed in Y for each ξ ≤ λ. Put Zξ = Z0 ∪ (Aμ × F<ξ ) for each ξ with 0 < ξ ≤ λ. We have Zλ = Aμ × Y since F<λ = F = Y . And {Zξ : ξ ≤ λ} is an increasing closed cover of Aμ × Y such that Zξ = ζ<ξ Zζ for each limit ordinal ξ ≤ λ. For each ξ < λ, there are a Qξ ∈ Q with Fξ ⊂ Qξ , an open neighborhood Vξ of μ in Aμ with Vξ × Qξ ⊂ W0 , and a νξ with −1 ≤ νξ < μ such that A ∩ (νξ , μ] ⊂ Vξ . Then (∗) Zξ+1 \ W is a D-space for each open set W in Aμ × Y with Zξ ∪ W0 ⊂ W . In fact, we should notice that Zξ+1 \ W ⊂ (Zξ+1 \ Zξ ) \ W0 ⊂ (Aμ × (F<(ξ+1) \ F<ξ )) \ (Vξ × Qξ ) ⊂ (Aμ × Fξ ) \ ((A ∩ (νξ , μ]) × Fξ ) = (Aμ \ (νξ , μ]) × Fξ ⊂ Aνξ × Y. By inductive hypothesis, Aνξ × Y is a D-space. Since Zξ+1 \ W is closed in Aμ × Y , it is also closed in Aνξ × Y . Hence (∗) is true. By induction on ξ ≤ λ, we will define an increasing sequence {Dξ : ξ ≤ λ} of closed discrete subsets in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ holds for each ξ ≤ λ, where Wξ = {ϕ(d) : d ∈ Dξ }, and Wζ ∩ Dξ ⊂ Dζ holds for each ζ, ξ with ζ < ξ ≤ λ. After finishing induction, we obtain a required closed and discrete set D in Aμ × Y by putting D = Dλ since Aμ × Y = Zλ ⊂ Wλ = {ϕ(d) : d ∈ Dλ } ⊂ Aμ × Y , thus we have Aμ × Y = {ϕ(d) : d ∈ Dλ } = {ϕ(d) : d ∈ D}. Let ξ ≤ λ and assume that an increasing sequence {Dζ : ζ ∈ ξ} of closed discrete subsets in Aμ × Y is defined, Dζ ⊂ Zζ ⊂ Wζ holds for each ζ ∈ ξ, and Wη ∩ Dζ ⊂ Dη holds for each η, ζ with η < ζ < ξ. We will find a closed discrete subset Dξ in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ holds, and Dζ ⊂ Dξ and Wζ ∩ Dξ ⊂ Dζ hold for each ζ < ξ. Case 2-1. ξ = 0. Dξ = D0 is already defined and satisfies the required condition. Case 2-2. ξ = ζ + 1 for some ζ < λ. It suffices to take a closed discrete subset Dξ in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ , Dζ ⊂ Dξ and Wζ ∩ Dξ ⊂ Dζ hold. If such Dξ is taken, then it also holds for each η < ζ that Dη ⊂ Dξ and Wη ∩ Dξ ⊂ Dη since Dη ⊂ Dζ ⊂ Dξ and Wη ∩ Dξ = Wη ∩ Wζ ∩ Dξ ⊂ Wη ∩ Dζ ⊂ Dη hold by inductive hypothesis.
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A closed discrete set Dζ in Aμ × Y is already defined and D0 ⊂ Dζ and Zζ ⊂ Wζ hold, so Wζ = {ϕ(d) : d ∈ Dζ } is an open set in Aμ × Y with Zζ ∪ W0 ⊂ Wζ . By the above (∗), Zζ+1 \ Wζ is a D-space. Take a closed discrete set Eζ in Zζ+1 \ Wζ (also in Aμ × Y ) such that Zζ+1 \ Wζ ⊂ {ϕ(e) : e ∈ Eζ }. Define Dξ by putting Dξ = Dζ+1 = Dζ ∪ Eζ . Then Dξ is a closed discrete set in Aμ × Y since so are Dζ and Eζ . Obviously Dζ ⊂ Dξ holds, and we have Wζ ∩ Dξ ⊂ Dζ since Wζ ∩ Eζ = ∅. By Dζ ⊂ Zζ ⊂ Zξ , Eζ ⊂ Zζ+1 = Zξ , Wζ = {ϕ(d) : d ∈ Dζ } ⊂ {ϕ(d) : d ∈ Dξ } = Wξ and Zξ \ Wζ = Zζ+1 \ Wζ ⊂ {ϕ(e) : e ∈ Eζ } ⊂ {ϕ(e) : e ∈ Dξ } = Wξ , it follows that Dξ ⊂ Zξ ⊂ Wξ . Case 2-3. ξ is a limit ordinal. Put Dξ = ζ<ξ Dζ . By Zξ = ζ<ξ Zζ and Wξ = {ϕ(d) : d ∈ Dξ } = ζ<ξ ( {ϕ(d) : d ∈ Dζ }) = ζ<ξ Wζ , it follows from the inductive hypothesis that Dζ ⊂ Zζ ⊂ Zξ and Zζ ⊂ Wζ ⊂ Wξ hold for each ζ < ξ, and Wζ ∩ Dη ⊂ Wζ ∩ Dmax{ζ,η}+1 ⊂ Dζ for each ζ, η < ξ. Therefore Dξ ⊂ Zξ ⊂ Wξ holds, and Wζ ∩ Dξ ⊂ Dζ holds for each ζ < ξ. To see that Dξ is closed discrete in Aμ × Y , let z ∈ Aμ × Y . If z ∈ Wξ , then take a ζ ∈ ξ with z ∈ Wζ . Since Dζ is closed and discrete, there is an open neighborhood U of z with U ⊂ Wζ and |U ∩ Dζ | ≤ 1. By U ∩ Dξ ⊂ U ∩ Wζ ∩ Dξ ⊂ U ∩ Dζ , we have |U ∩ Dξ | ≤ 1. If z ∈ / Wξ , then (Aμ × Y ) \ Zξ is an open neighborhood of z which is disjoint from Dξ since Dξ ⊂ Zξ ⊂ Wξ and Zξ is a closed set in Aμ × Y . Hence Dξ is a closed discrete set in Aμ × Y . 2
Recall that every subspace of an ordinal is a GO-space. Douwen and Lutzer proved the following remarkable result. Theorem 5.3. ([4, Theorem 1.2]) A GO-space X is paracompact if and only if it is a D-space. By Proposition 5.2 and Theorem 5.3, we immediately have Corollary 5.4. Let A be a non-empty subspace of an ordinal, and Y a non-empty submetacompact space. Then A × Y is a D-space if and only if both A and Y are D-spaces. Remark 5.5. Each subspace of an ordinal is scattered, in particular, C-scattered. However, there is a paracompact subspace A of ω1 + 1 which is neither compact nor hereditarily submetacompact. In fact, let A = (ω1 + 1) \ {ω}. Since A satisfies Lemma 5.1(c), it is paracompact. Let B = ω1 \ {ω}. Then B is stationary in ω1 . By Lemma 5.1(c), B is not paracompact. Since paracompactness and submetacompactness are equivalent in the class of GO-spaces, the subspace B in A is not submetacompact. So Proposition 5.2 cannot be derived from Lemma 3.4, Corollary 3.6 or Proposition 4.5. 6. Countable products of subparacompact DC-like spaces Let m ∈ ω. A subset of the form i≤m Fi in a finite product i≤m Xi of spaces Xi , i ≤ m, is a closed rectangle if Fi is closed in Xi for each i ≤ m. Let X = i∈ω Xi be a countable product of spaces Xi , i ∈ ω. A subset of the form F = i∈ω Fi in it is called a closed (compact) box in X if Fi is a closed (compact) set in Xi for each i ∈ ω. Let and ≺ be binary relations on a set P . In particular, recall that is a partial order on P if • p p for each p ∈ P , • for each p, q, r ∈ P , if p q and q r, then p r.
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We call {pk : k ∈ ω} ⊂ P a -increasing sequence if pk pk+1 for each k ∈ ω, and we call {pk : k ∈ ω} ⊂ P a -decreasing sequence (≺-decreasing sequence) if pk+1 pk (pk+1 ≺ pk ) for each k ∈ ω. If is a partial order on P , then it is obvious that {pk : k ∈ ω} ⊂ P is a -increasing ( -decreasing) sequence if and only if pj pk (pk pj ) for every j ≤ k < ω. For Q ⊂ P , we call q a ≺-minimal element of Q if q ∈ Q and there is no p ∈ Q with p ≺ q. The following is our main result proved by using an elementary submodel. Theorem 6.1. If Xi is a subparacompact DC-like space for each i ∈ ω, then
i∈ω
Xi is a D-space.
Proof. Let X = i∈ω Xi , and let D X be the family of all closed discrete sets in X. Let ϕ be an open neighborhood assignment for X. Put ϕ[D] = {ϕ(d) : d ∈ D} for each D ⊂ X. To see that X is a D-space, it suffices to find a D ∈ D X with X = ϕ[D]. For D, D ∈ D X , let D < D denote that D ⊂ D and D ∩ ϕ[D] ⊂ D. Then < is a partial order on D X . Let Ψ denote the set of all functions ψ : D X → D X such that D < ψ(D) for every D ∈ D X . Let θ be a sufficiently large regular cardinal, and M a countable elementary submodel of Hθ such that X, τ (X), ϕ ∈ M, where recall that τ (X) is the topology on X. Let M ∩ Ψ = {ψk : k ∈ ω} and |{k ∈ ω : ψk = ψ}| = ω for every ψ ∈ M ∩ Ψ. By induction on k ∈ ω, we define a <-increasing sequence {Dk : k ∈ ω} ⊂ M ∩ D X by putting D0 = ∅ and Dk+1 = ψk (Dk ) for each k ∈ ω. After finishing induction, put Dω = k∈ω Dk . We will show that Dω ∈ D X and X = ϕ[Dω ]. The former is implied by the latter. Actually, for each x ∈ X, we can take j ∈ ω with x ∈ ϕ[Dj ]. Since Dj ∈ D X , there is an open neighborhood V of x such that V ⊂ ϕ[Dj ] and |V ∩Dj | ≤ 1. For each k with j < k < ω, since Dj < Dk , we have V ∩Dk ⊂ ϕ[Dj ] ∩Dk ⊂ Dj , so V ∩ Dω = j
Since M is an elementary submodel, each L ∈ M ∩ σL is expressed as L = n ∈ ω} ⊂ M ∩ L, so we have (M ∩ σL) = (M ∩ L). Put 2 F⊂ [x, M] = {E, F ∈
(M ∩ L) : x ∈ E} = {E, F ∈
n∈ω
Ln for some {Ln :
(M ∩ σL) : x ∈ E}.
2 2 Since X, X ∈ F⊂ and {X, X } ∈ M ∩ L, we have X, X ∈ F⊂ [x, M].
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2 Define a partial order ≤2 and a binary relation <2i , for each i ∈ ω, on F⊂ by
• E , F ≤2 E, F if and only if E ⊂ E and F ≤s F , • E , F <2i E, F if and only if E ⊂ E and F
˙ F˙ ∈ F 2 [x, M] with F˙ ≤s X and i ∈ ω. Then there is a <2 -minimal element E, ¨ F¨ of Claim 2. Let E, ⊂ i 2 2 ¨ ˙ ¨ ¨ ˙ ¨ F⊂ [x, M] such that E, F ≤ E, F and Ei ∩ si (Fi ) is compact. 2 Proof. By Claim 1, there is not a <2i -decreasing sequence {E (j) , F (j) : j ∈ ω} ⊂ F⊂ [x, M] starting from (0) (0) 2 2 ˙ F˙ . So there is a < -minimal element E , F of F⊂ [x, M] with E , F ≤2 E, ˙ F˙ . E , F = E, i ¨ Take L ∈ M ∩ L with E , F ∈ L . Then there is an L satisfying:
2 (··) L¨ ∈ σL and L¨ ⊂ {E , F ∈ F⊂ : Ei ∩ si (Fi ) is compact} hold. And for each E, F ∈ L , E ⊂ {E : ¨ E , F ≤2 E, F } holds. E , F ∈ L, To find an L¨ satisfying (··), let E, F be an arbitrary member of L . Since Xi is assumed to be subparacompact and si (Fi ) is the union of a discrete collection of compact subsets in Xi , we see that Ei is covered by a σ-discrete subcollection C of {C ⊂ Ei : C is a non-empty closed set in Xi such that C ∩ si (Fi ) is compact}. ¨ ¨ F¨ such that E ¨ is obtained by replacing the i-factor Ei of E with some Let L(E, F ) be the family of all E, ¨ C ∈ C, and F¨ = F . By putting L¨ = {L(E, F ) : E, F ∈ L }, we obtain an L¨ satisfying (··). Since the condition (··) can be expressed by a formula with parameters X, τ, s, i, L ∈ M, we can take ¨ Since E , F ∈ L and x ∈ E , L¨ ∈ M satisfying (··) (though it may be different from the old L). ¨ F¨ ∈ L¨ with E, ¨ F¨ ≤2 E , F ≤2 E, ˙ F˙ and x ∈ E. ¨ Since L¨ ∈ M ∩ σL, we have we obtain E, 2 2 2 ¨ ¨ F¨ is also a ¨ E, F ∈ F⊂ [x, M]. Since E , F is a
i≤m
Xi with the projection π≤m : X → X≤m .
2 Claim 3. There are an E ∗ , F ∗ ∈ F⊂ [x, M], an m ∈ ω, and a finite subfamily L∗ of
2 {E ∗∗ , F ∗∗ ∈ F⊂ : E ∗∗ , F ∗∗ <2i E ∗ , F ∗ }
i≤m
such that 2 (i) E ∗ , F ∗ is a <2i -minimal element of F⊂ [x, M] for each i ≤ m, (ii) for each k ∈ ω with m ≤ k, there is a Dk∗ ∈ D X such that Dk∗ ⊂ Dk ∪ E ∗ , Dk < Dk∗ and E ∗ ⊂ ϕ[Dk∗ ] ∪ E[L∗ ] . 2 Proof. By Claim 2, we can take a ≤2 -decreasing sequence {E (i) , F (i) : i ∈ {−1} ∪ ω} ⊂ F⊂ [x, M] starting (−1) (−1) (i) (i) 2 from E ,F = X, X such that for each i ∈ ω the pair E , F is a
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• Vy ⊂ ϕ(y), • if y ∈ ϕ[Dω ], then Vy ⊂ ϕ[Dky ] for some ky ∈ ω, • if y ∈ / i∈ω E (i) , then Vy ∩ E (iy ) = ∅ for some iy ∈ ω. −1 We may assume that Vy = π≤m (Uy ) for an my ∈ ω and an open set Uy in X≤my . Since K is compact, y there is a finite subset J of K such that K ⊂ y∈J Vy . Take an m ∈ ω such that my ≤ m for each y ∈ J, ky ≤ m for each y ∈ J ∩ ϕ[Dω ], iy ≤ m for each y ∈ J \ i∈ω E (i) , and if Kj = ∅ for some j ∈ ω, then j ≤ m for the least such j. We show that E ∗ , F ∗ = E (m) , F (m) satisfies the required condition. For each i ≤ m, since E (i) , F (i) 2 [x, M] and E ∗ , F ∗ ≤2 E (i) , F (i) , we see that F ∗ i = F (i) i and E ∗ , F ∗ is is a <2i -minimal element of F⊂ ∗ also a <2i -minimal element of F⊂ [x, M]. Put
U=
{u ∈ X≤m : u {0, · · · , my } ∈ Uy },
y∈J −1 and put V = π≤m (U ). Then U and V are open sets in X≤m and X, respectively, such that K≤m := Vy . So X≤m has a finite cover by closed rectangles which are contained in U i≤m Ki ⊂ U and V = y∈J or disjoint from K≤m . And if i≤m Ei is a closed rectangle in X≤m which is disjoint from K≤m , then there is an i0 ≤ m such that Ei0 ∩Ki0 = ∅, so Ei0 ∩E ∗ i0 ⊂ Ei0 ∩E (i0 ) i0 ⊂ F (i0 ) i0 \si0 (F (i0 ) i0 ) = F ∗ i0 \si0 (F ∗ i0 ). If E ∗∗ ∈ F is obtained from E ∗ by replacing i-factor E ∗ i with Ei ∩E ∗ i for each i ≤ m, and F ∗∗ ∈ F is obtained from F ∗ by replacing the i0 -factor Fi∗0 with Ei0 ∩ Ei∗0 , then E ∗∗ , F ∗∗ <2i0 E ∗ , F ∗ holds. Hence there is a 2 finite subfamily L∗ of i≤m {E ∗∗ , F ∗∗ ∈ F⊂ : E ∗∗ , F ∗∗ <2i E ∗ , F ∗ } such that E ∗ ⊂ V ∪ E[L∗ ] . Put D∗ = J ∩( i∈ω E (i) ) \ϕ[Dω ]. Put Dk∗ = Dk ∪D∗ for each k ∈ ω with m ≤ k. We have Dk∗ ∈ D X since D∗ is a finite set. By D∗ ⊂ i∈ω E (i) ⊂ E (m) = E ∗ , we have Dk∗ ⊂ Dk ∪E ∗ . By ϕ[Dk ] ∩D∗ ⊂ ϕ[Dω ] ∩D∗ = ∅, we have ϕ[Dk ] ∩ Dk∗ ⊂ Dk , therefore Dk < Dk∗ holds. Let y ∈ J. If y ∈ i∈ω E (i) and y ∈ / ϕ[Dω ], then y ∈ D∗ ⊂ Dk∗ , so Vy ∩ E ∗ ⊂ Vy ⊂ ϕ(y) ⊂ ϕ[Dk∗ ]. If y ∈ ϕ[Dω ], then since ky ≤ m ≤ k, we have Dky ⊂ Dk ⊂ Dk∗ , so Vy ∩ E ∗ ⊂ Vy ⊂ ϕ[Dky ] ⊂ ϕ[Dk∗ ]. If y ∈ / i∈ω E (i) , then since iy ≤ m, we have E ∗ = E (m) ⊂ E (iy ) , so Vy ∩ E ∗ ⊂ Vy ∩ E (iy ) = ∅ ⊂ ϕ[Dk∗ ]. Hence, V ∩ E ∗ = y∈J (Vy ∩ E ∗ ) ⊂ ϕ[Dk∗ ]. By E ∗ ⊂ V ∪ E[L∗ ], we have E ∗ ⊂ ϕ[Dk∗ ] ∪ E[L∗ ] . 2
Take E ∗ , F ∗ , m and L∗ described in Claim 3. And take an L ∈ M ∩ L such that E ∗ , F ∗ ∈ L. Claim 4. There is a ψ such that: (∗) ψ ∈ Ψ holds. And for each D ∈ D X and D = ψ(D), there is an L such that: (∗∗) L ∈ L holds. And for each E, F ∈ L, if (†) there are a D† ∈ D X and a finite subfamily L† of
2 {E † , F † ∈ F⊂ : E † , F † <2i E, F }
i≤m
such that D† ⊂ D ∪ E, D < D† and E ⊂ ϕ[D† ] ∪ E[L† ] , then E ⊂ ϕ[D ] ∪ i≤m {E : E , F ∈ L , E , F <2i E, F }. Proof. It suffices to show that for each D ∈ D X , there are a D ∈ D X with D < D and an L satisfying the condition (∗∗). Let L0 be the family of all E, F ∈ L satisfying the condition (†). If L0 = ∅, then D = D and L = ∅ satisfy the required condition. So assume that L0 = ∅. For each E, F ∈ L0 , fix a † † † † D(E,F ) , L(E,F ) = D , L witnessing (†). Put
Y. Hirata, Y. Yajima / Topology and its Applications 195 (2015) 297–311
D =
† {D(E,F ) : E, F ∈ L0 } and L =
307
{L†(E,F ) : E, F ∈ L0 }.
† † Let E, F ∈ L0 . Then D(E,F ) \D ⊂ E and ϕ[D] ∩D(E,F ) ⊂ D. By L ∈ L, E[L] = {E : E, F ∈ L} is locally † finite in X, so we have D ∈ D X and L ∈ L. And ϕ[D] ∩D = {ϕ[D] ∩D(E,F ) : E, F ∈ L0 } ⊂ D holds, so † † we have D < D . By the definition, it is obvious that E ⊂ ϕ[D(E,F ) ] ∪ E[L(E,F ) ] ⊂ ϕ[D ] ∪ i≤m {E : E , F ∈ L , E , F <2i E, F } holds for each E, F ∈ L satisfying (†). 2
Since the condition (∗) in Claim 4 can be expressed by a formula with parameters X, τ, s, m, ϕ, L ∈ M, we can take ψ ∈ M satisfying (∗) (though it may be different from the old ψ). Take a k ∈ ω with ψk = ψ and m ≤ k. Put D = Dk and D = Dk+1 = ψk [Dk ] = ψ[D]. By the definition of ψ, there is an L satisfying the condition (∗∗) in Claim 4. Since the condition (∗∗) can be expressed by a formula with parameters X, τ, s, m, ϕ, L, D, D ∈ M, we can take L ∈ M satisfying (∗∗). Then D† = Dk∗ and L† = L∗ in Claim 3 2 witness that (†) holds for E, F = E ∗ , F ∗ . It follows from E ∗ , F ∗ ∈ F⊂ [x, M] that x ∈ E ∗ ⊂ ϕ[D ] ∪
{E : E , F ∈ L , E , F <2i E ∗ , F ∗ }.
i≤m
2 By L ∈ M ∩L, we have E , F ∈ (M ∩L) for each E , F ∈ L . On the other hand, E , F ∈ / F⊂ [x, M] 2 ∗ ∗ ∗ ∗ 2 2 if E , F
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cs,t ∈ D(s, t) with cs,t = ds,t and γ(s, t) ∈ ω1 with cs,t ∈ Fγ(s,t) . If D(s, t) = ∅, then put α(s, t) = 0. If |D(s, t)| = 1, then put α(s, t) = δ(s, t). And if |D(s, t)| ≥ 2, then put α(s, t) = max{δ(s, t), γ(s, t)}. Inductively, we take a strictly increasing sequence {βn : n ∈ ω} ⊂ ω1 . For n = 0, put β0 = 0. Let n ∈ ω and assume that βn ∈ ω1 is defined. Take βn+1 ∈ ω1 such that βn < βn+1 and α(s, t) < βn+1 for each finite subset s of βn and for each t ∈ N s . After finishing induction, let β = sup{βn : n ∈ ω}. Take y ∈ Fβ and d ∈ D with y ∈ V (d). Take the α ∈ ω1 with d ∈ Fα . We have y(α) = 0 by y ∈ V (d), so β ≤ α holds by y ∈ Fβ . Hence there is a z ∈ Fβ with z β = d β. Let W be an arbitrary open neighborhood of z in F . Then there is a finite subset sˆ of ω1 such that {x ∈ F : x sˆ = z sˆ} ⊂ W . Put s = sˆ ∩ β and t = d s. Then D(s, t) is non-empty since d ∈ D(s, t). So ds,t ∈ D(s, t) and δ(s, t) ∈ ω1 with ds,t ∈ Fδ(s,t) are defined. Take an m ∈ ω with s ⊂ βm . We have δ(s, t) ≤ α(s, t) < βm+1 < β ≤ α. In particular, δ(s, t) = α, so Fδ(s,t) ∩ Fα = ∅, thus ds,t = d. Hence, |D(s, t)| ≥ 2. So cs,t ∈ D(s, t) with cs,t = ds,t and γ(s, t) ∈ ω1 with c(s, t) ∈ Fγ(s,t) are defined. And we have γ(s, t) ≤ α(s, t) < βm+1 < β. Since ds,t , cs,t ∈ D(s, t) and s ⊂ β, we have ds,t s = cs,t s = t = d s = z s. For each ξ ∈ sˆ \ s, since δ(s, t), γ(s, t) < β ≤ ξ, ds,t ∈ Fδ(s,t) , cs,t ∈ Fγ(s,t) and z ∈ Fβ , we have ds,t (ξ) = cs,t (ξ) = 0 = z(ξ). Therefore ds,t sˆ = cs,t sˆ = z sˆ holds, thus ds,t , cs,t ∈ D ∩ W . We see that D ∩ W has at least two members for each open neighborhood W of z, hence D is not a closed discrete subset in F . Next, we will show that F is dually discrete (this case seems to be more difficult than the former). Let ϕ be an open neighborhood assignment for F . We would like to find a discrete subset D in F with {ϕ(d) : d ∈ D} = F . For each x ∈ F , take and fix a finite subset sx of ω1 such that {y ∈ F : y sx = x sx } ⊂ ϕ(x). Let δ ≤ ω1 . Set S(δ) = {s, t : s is a finite subset of δ, t ∈ N s }. For each s, t ∈ S(ω1 ), put Fα (s, t) = {x ∈ Fα : sx ∩α = s, x s = t} and A(s, t) = {α ∈ ω1 : Fα (s, t) = ∅}. And set S0 (δ) = {s, t ∈ S(δ) : A(s, t) is bounded in ω1 }, S1 (δ) = {s, t ∈ S(δ) : A(s, t) is unbounded in ω1 }. We denote S(ω1 ), S0 (ω1 ) and S1 (ω1 ) by S, S0 and S1 , respectively. Take a ξ ∈ ω1 . For each s, t ∈ S(ξ) = S0 (ξ) ∪ S1 (ξ), take a βξ (s, t) ∈ ω1 such that A(s, t) ⊂ βξ (s, t) and ξ < βξ (s, t) if s, t ∈ S0 (ξ) and that βξ (s, t) ∈ A(s, t) ∩ (ξ, ω1 ) if s, t ∈ S1 (ξ). Moreover, take a βξ ∈ ω1 with βξ (s, t) < βξ for each s, t ∈ S(ξ). Put C = {δ ∈ ω1 : δ is a limit ordinal such that ξ < δ implies βξ < δ}. Then C is a closed unbounded set in ω1 . It is easily verified that for each δ ∈ C, • A(s, t) ⊂ δ holds for each s, t ∈ S0 (δ), • A(s, t) ∩ (γ, δ) = ∅ holds for each s, t ∈ S1 (δ) and each γ with −1 ≤ γ < δ. And take a pairwise disjoint family {C(s, t) : s, t ∈ S1 } of unbounded subsets in ω1 such that C(s, t) ⊂ {γ ∈ C : s ⊂ γ} for each s, t ∈ S1 . Let γ ∈ C ∪ {−1}. Put δγ = min{δ ∈ C : γ < δ} and F ∗ [γ] = γ<α<δγ Fα . By enumerating the elements, let S(δγ ) = {sγ,n , tγ,n : n ∈ ω}. We inductively define an increasing sequence {Dγ,n : n ∈ ω} of finite subsets in F ∗ [γ]. For n = 0: If γ ∈ C(sγ , tγ ) for some sγ , tγ ∈ S1 , then by δγ ∈ C, γ < δγ and sγ ⊂ γ ⊂ δγ , we have sγ , tγ ∈ S1 (δγ ), and so we can take an αγ ∈ A(sγ , tγ ) with γ < αγ < δγ . By Fαγ (sγ , tγ ) = ∅, we obtain a
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309
dγ ∈ Fαγ (sγ , tγ ). Then dγ ∈ Fαγ ⊂ F ∗ [γ] holds. Define Dγ,0 by putting Dγ,0 = {dγ }. If γ ∈ / C(s, t) for any s, t ∈ S1 , then let Dγ,0 = ∅. For n + 1: Let n ∈ ω and assume that a finite subset Dγ,n of F ∗ [γ] is defined. If there is an x ∈ F ∗ [γ] \ {ϕ(d) : d ∈ Dγ,n } such that sx ∩ δγ = sγ,n and x sγ,n = tγ,n , then take and fix one of such dγ,n = x and let Dγ,n+1 = Dγ,n ∪ {dγ,n }. If there is no such x, then let Dγ,n+1 = Dγ,n . After finishing induction, let Dγ = n∈ω Dγ,n . By the definition, Dγ is a discrete subset in F ∗ [γ]. Let D = γ∈C∪{−1} Dγ . We show that D is discrete. Since Dγ is a discrete subset in F ∗ [γ] for each γ, it suffices to show that {F ∗ [γ] : γ ∈ C ∪ {−1}} is a discrete collection in F ∗ := γ∈C∪{−1} F ∗ [γ]. Let x ∈ F ∗ [γ] for some γ ∈ C ∪ {−1}. Then x ∈ Fα for some α with γ < α < δγ . By the definition of Fα , we have x(γ) = 0 and x(α) = 0. We obtain an open neighborhood V := {y ∈ F : y(γ) = 0, y(α) = 0} of x which is disjoint from F ∗ [γ ] for any γ ∈ C ∪ {−1} with γ = γ. Actually, let x ∈ F ∗ [γ ]. Then x ∈ Fα for some α with γ < α < δγ . If γ < γ , then α < δγ ≤ γ < α by the minimality of δγ , so we have x (α) = 0. If γ > γ , then γ ≥ δγ > α by the minimality of δγ , so we have x (γ) = 0. In any case, we obtain x ∈ / V. To see that F is dually discrete, it suffices to show that {ϕ(d) : d ∈ D} = F holds. Let x ∈ F . We would like to find a d ∈ D with x ∈ ϕ(d). Take the α ∈ ω1 with x ∈ Fα . And let s = sx ∩ α and t = x s. Then s, t ∈ S(α) = S0 (α) ∪ S1 (α). In case s, t ∈ S0 (α): By x ∈ Fα (s, t), we have α ∈ A(s, t) \ α, thus A(s, t) ⊂ α. Hence, α ∈ / C and so γ < α < δγ for some γ ∈ C ∪ {−1}, that is x ∈ F ∗ [γ]. Take an m ∈ ω with sγ,m = sx ∩ δγ and tγ,m = x sγ,m . If x ∈ {ϕ(d) : d ∈ Dγ,m }, then we obtain a d ∈ Dγ,m ⊂ Dγ ⊂ D with x ∈ ϕ(d), and such d is a required one. Otherwise, dγ,m ∈ Dγ,m+1 ⊂ Dγ ⊂ D is defined. Let d = dγ,m . By the definition, we have d ∈ F ∗ [γ], sd ∩ δγ = sγ,m and x sγ,m = tγ,m = d sγ,m . There is an α with γ < α < δγ and d ∈ Fα . For each ξ ∈ sd \ δγ , it follows from x ∈ Fα , d ∈ Fα and α, α < δγ ≤ ξ that x(ξ) = d(ξ) = 0. Therefore, x sd = d sd holds, thus x ∈ ϕ(d). Hence d is a required one. In case s, t ∈ S1 (α): Take a γ ∈ C(s, t) with α ≤ γ. Then sγ , tγ = s, t , an ordinal αγ with γ < αγ < δγ , and a dγ ∈ Fαγ (s, t) are defined. Let d = dγ . We have d ∈ Dγ,0 ⊂ Dγ ⊂ D. It follows from d ∈ Fαγ (s, t) that sd ∩ αγ = s and x s = t = d s. And for each ξ ∈ sd \ αγ , it follows from x ∈ Fα , d ∈ Fαγ and α ≤ γ < αγ ≤ ξ that x(ξ) = d(ξ) = 0. Therefore, x sd = d sd , thus x ∈ ϕ(d) holds. Hence d is a required one. 2 Since every closed subspace of a D-space is also a D-space, we immediately have Corollary 7.2. The space N ω1 is not a D-space. Moreover, combining this with Theorem 6.1, the following is obtained. Corollary 7.3. Let λ∈Λ Xλ be an uncountable product of non-empty subparacompact DC-like spaces. Then λ∈Λ Xλ is a D-space if and only if each Xλ is compact except countably many λ ∈ Λ. Proof. The “if” part follows from Lemma 3.4 and Theorem 6.1. Conversely, assume that there is Γ ⊂ Λ such that |Γ| = ω1 and Xλ is not compact for each λ ∈ Γ. Since each Xλ is subparacompact, such an Xλ is not countably compact. So there is an infinite countable closed discrete subset Nλ in Xλ for each λ ∈ Γ. Then to a closed subspace in λ∈Λ Xλ . It follows from Corollary 7.2 that λ∈Γ Nλ is λ∈Γ Nλ is homeomorphic not a D-space. Hence λ∈Λ Xλ is not a D-space. 2 Since each countably compact and submetacompact space is compact, the following is similarly obtained. Corollary 7.4. Let X be a submetacompact space and κ an uncountable cardinal. Then X κ is a D-space if and only if X is compact.
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8. Problems Since subparacompactness implies submetacompactness, it is natural from Corollary 3.6 to raise Problem 1. Let X be a DC-like space and Y a D-space. If X is subparacompact and Y is submetacompact, or if X is submetacompact and Y is subparacompact, is then X × Y a D-space? From the negative aspect of this, we raise Problem 2. Are there a DC-like space X and a D-space Y such that X × Y is not a D-space? Remark 8.1. Every binary open cover of a normal and rectangular product X × Y has a σ-locally finite refinement by closed rectangles. Using this property instead of D-products, we can show the following: Let X be a DC-like space and Y a D-space. If X × Y is normal and rectangular, then it is a D-space. Since subparacompact C-scattered spaces are DC-like, it is natural from Proposition 4.5 to ask Problem 3. Let X be a submetacompact C-scattered space and Y a submetacompact D-space. Is then X ×Y a D-space? Since every subspace of an ordinal is a GO-space, it is natural from Proposition 5.2 to ask Problem 4. If X and Y are paracompact GO-spaces, is then X × Y D-space? As well as Problem 1, it is natural from Theorem 6.1 to raise Problem 5. Let Xi be a submetacompact DC-like (or C-scattered) space for each i ∈ ω. Is then D-space?
i∈ω
Xi a
Remark 8.2. We cannot expect an analogical result to Theorem 3.1 for countable products. In fact, consider the irrational space N ω . It was shown in [6] that N ω is not totally paracompact (i.e., it has a base which has no locally finite subcover). Obviously, N is a 1-like space. Since it follows from [21, Theorem 5.1] that every paracompact DC-like space is totally paracompact. Hence N ω is not DC-like. In particular, N ω is not D1-like. Finally, it is natural to raise the following problem from the previous section. Problem 6. Is the space N ω1 dually discrete? References [1] O.T. Alas, L.R. Junqueira, G. Wilson, Dually discrete spaces, Topol. Appl. 155 (2008) 1420–1425. [2] C.R. Borges, A.C. Wehrly, A study of D-spaces, Topol. Proc. 16 (1991) 7–15. [3] J. Chaber, G. Gruenhage, R. Pol, On Souslin sets and embeddings in integer-valued function spaces on ω1 , Topol. Appl. 82 (1998) 71–104. [4] E.K. van Douwen, D.J. Lutzer, A note on paracompactness in generalized ordered spaces, Proc. Am. Math. Soc. 125 (1997) 1237–1245. [5] E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pac. J. Math. 81 (1979) 371–377. [6] B. Fitzpatrick Jr., R.M. Ford, On the equivalence of small and large inductive dimension in certain metric spaces, Duke Math. J. 34 (1967) 33–37. [7] F. Galvin, R. Telgársky, Stationary strategies in topological games, Topol. Appl. 22 (1986) 51–69. [8] G. Gruenhage, A survey of D-spaces, Contemp. Math. 533 (2011) 13–28. [9] H. Guo, H. Junnila, D-spaces and thick covers, Topol. Appl. 158 (2011) 2111–2121.
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H.J.K. Junnila, On submetacompactness, Topol. Proc. 3 (1978) 375–405. J.C. Martínez, On finite unions and finite products with the D-property, Topol. Appl. 156 (2011) 223–228. L. -Xue Peng, On some sufficiencies of D-spaces, J. Beijing Inst. Technol. 16 (1996) 229–233. L. -Xue Peng, On products of certain D-spaces, Houst. J. Math. 34 (2008) 165–179. L. -Xue Peng, On linear neighborhood assignments and dually discrete spaces, Topol. Appl. 155 (2008) 1867–1874. B.M. Scott, More about orthocompactness, Topol. Proc. 5 (1980) 155–184. A.H. Stone, Paracompactness and product spaces, Bull. Am. Math. Soc. 54 (1948) 977–982. P. Szeptycki, Y. Yajima, Covering properties, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Atlantis Press, 2014, pp. 801–824. H. Tanaka, Covering properties in Cartesian products, Tsukuba J. Math. 17 (1993) 565–587. R. Telgársky, C-scatteded and paracompact spaces, Fundam. Math. 73 (1971) 59–74. R. Telgársky, Spaces defined by topological games, Fundam. Math. 88 (1975) 193–223. Y. Yajima, Topological games and products II, Fundam. Math. 117 (1983) 47–60. Y. Yajima, Topological games and products III, Fundam. Math. 117 (1983) 223–238. Y. Yajima, Topological games and products, in: K. Morita, J. Nagata (Eds.), Topics in General Topology, North-Holland, 1989, pp. 523–562.
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Topology and its Applications www.elsevier.com/locate/topol
On the D-property of certain products ✩ Yasushi Hirata, Yukinobu Yajima ∗ Department of Mathematics, Kanagawa University, Yokohama, 221-8686 Japan
a r t i c l e
i n f o
Article history: Received 18 May 2014 Received in revised form 2 September 2014 Available online 2 October 2015 MSC: primary 54B10, 54D20 secondary 03E10, 03E75
a b s t r a c t We first point out that the product of a subparacompact DC-like space and a subparacompact D-space is a D-space. As a main result, we use elementary submodels to prove that any countable product of subparacompact DC-like spaces is a D-space. More generally, we prove that any uncountable product of subparacompact DC-like spaces is a D-space if and only if all but countably many of the factors are compact. © 2015 Elsevier B.V. All rights reserved.
Keywords: D-space DC-like Subparacompact Submetacompact Elementary submodel
1. Introduction An open neighborhood assignment for a space X is a function ϕ : X → τ (X) such that x ∈ ϕ(x) for each x ∈ X, where τ (X) denotes the topology of X. A space X is called a D-space (or has the D-property) [5] if for each open neighborhood assignment ϕ for X, there is a closed discrete subset D in X with {ϕ(x) : x ∈ D} = X. The concept of D-spaces was introduced by van Douwen and Pfeffer [5] and they proved that every finite power of the Sorgenfrey line S is a D-space. Since then there have been very many papers devoted to the study of D-spaces. In particular, there is a survey by Gruenhage [8] which gives a nice overview of D-spaces. The D-property is a covering property. It is natural to consider whether given covering properties are productive. However, the D-property is not even finitely productive. In fact, it was shown in [1] that there are a Lindelöf D-space X and a separable metric space M such that X × M is not a D-space. On the other hand, Peng [13] proved that any countable product of paracompact DC-like spaces is a D-space. ✩
This research was supported by Grant-in-Aid for Scientific Research (C) 24540147.
* Corresponding author. E-mail addresses: [email protected] (Y. Hirata), [email protected] (Y. Yajima). http://dx.doi.org/10.1016/j.topol.2015.09.036 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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First, we point out that the product of a subparacompact DC-like space and a subparacompact D-space is a D-space. Second, we show that the product of a hereditarily submetacompact C-scattered space and an arbitrary D-space is a D-space. Third, we also show that the product of a paracompact subspace of an ordinal and a submetacompact D-space is a D-space The first result gives us motivation to extend Peng’s result above by replacing paracompactness with subparacompactness. Here it is proved by using an elementary submodel that every countable product of subparacompact DC-like spaces is a D-space. It has been suggested in [17] that elementary submodels will be very useful to study covering properties. This result may be considered as an illustration of this. Moreover, for the uncountable product case, we prove that the uncountable power of an infinite countable discrete space is not a D-space. Finally, we raise several unsolved problems. All spaces here are assumed to be regular T1 . We denote by ω the set of all natural numbers. 2. Topological games Let K be a non-void class of spaces. For such a class K, we denote by DK the class of all spaces which have a discrete (clopen) cover by members of K. For example, when 1 denotes the class of all one-point spaces, D1 means the class of all discrete spaces. As such a K, we mainly use the two classes C and D as follows: • C denotes the class of all compact spaces and • D denotes the class of all D-spaces. In 1975, Telgársky introduced and studied the topological game G(K, X). Definition 1. ([20]) Let X be a space. Let K be a class of spaces which is hereditary with respect closed sets (i.e., for each closed set F in Y ∈ K, F ∈ K holds). In the game G(K, X), two players choose closed subsets En and Fn in X for each n ∈ ω \ {0} in turn E1 , F1 , E2 , F2 , · · ·. Player I chooses En ∈ K with En ⊂ Fn−1 , where F0 = X. Player II chooses Fn with Fn ⊂ Fn−1 \ En . Player I wins if n∈ω Fn = ∅. Player II wins otherwise. A space X is said to be K-like if Player I has a winning strategy in the game G(K, X). Theorem 2.1. (Galvin and Telgársky [7]) A space X is K-like if and only if there is a function s : 2X → 2X ∩ K, where 2X denotes the family of all closed sets in X, such that (i) s(F ) ⊂ F for each closed set F in X, (ii) if {Fn }n∈ω is a decreasing sequence of closed sets in X satisfying that s(Fn−1 ) ∩ Fn = ∅ for each n ∈ ω, where F0 = X, then n∈ω Fn = ∅ holds. Such a function s above is called a stationary winning strategy for Player I in G(K, X). We can arbitrarily take the first choice of Player I for the s as follows. Proposition 2.2. Let X be a K-like space with a non-empty closed set E in X with E ∈ K. Then there is a stationary winning strategy s for Player I in G(K, X) such that (iii) s(X) = E, (iv) s(F ) is non-empty for each F ∈ 2X \ {∅}. Proof. By Theorem 2.1, we can take a stationary winning strategy s0 for Player I in G(K, X). We define a function s : 2X → 2X ∩ K as follows; for each F ∈ 2X meeting E, let s(F ) = E ∩ F , and for each F ∈ 2X
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disjoint from E, let s(F ) = s0 (F ) if F ∩ s0 (X) = ∅ and let s(F ) = F ∩ s0 (X) if F ∩ s0 (X) = ∅. Then s satisfies (iii) and (iv). Let s(X), F1 , s(F1 ), F2 , s(F2 ), · · · be a play in G(K, X). When F1 ∩ s0 (X) = ∅, s0 (X), F2 , s0 (F2 ), F3 , s0 (F3 ), · · · is a play in G(K, X). Hence we have n∈ω Fn = n≥2 Fn = ∅. When F1 ∩ s0 (X) = ∅, s0 (X), F1 , s0 (F1 ), F2 , s0 (F2 ), · · · is a play in G(K, X). Hence we have n∈ω Fn = ∅. Thus s is a stationary winning strategy for Player I in G(K, X). 2 3. Subparacompact DC-like spaces Let X × Y be a product of two spaces. A subset of the form E × F in X × Y is called a rectangle. A rectangle E × F is called a closed rectangle in X × Y if E and F are closed in X and Y , respectively. Definition 2. ([21]) A product X × Y of two spaces is called a D-product if for any disjoint pair E, F of closed sets E and F in X × Y , there is a σ-discrete collection F by closed rectangles in X × Y such that F ⊂ F ⊂ (X × Y ) \ E. For two classes K1 and K2 of spaces which are hereditary with respect closed sets, we put K1 × K2 = {E : E is a closed set in X × Y with X ∈ K1 and Y ∈ K2 }. The following result was proved in [21, Theorem 2.1] (see [23, Theorem 5.3] for its improved proof). Theorem 3.1. ([21]) Let X and Y be a K1 -like space and a K2 -like space, respectively. If X ×Y is a D-product, then it is D(K1 × K2 )-like. Recall that a space X is subparacompact if every open cover of X has a σ-locally finite (or σ-discrete) closed refinement. A typical example of D-products was given as follows. Proposition 3.2. ([22]) If X is a subparacompact DC-like space and Y is a subparacompact space, then X × Y is a subparacompact D-product. The following is due to Peng, which was also stated in [8, Theorem 3.3] with a proof. Lemma 3.3. ([12]) Every D-like space is a D-space. By DC ⊂ D, note that every DC-like space is a D-space. Lemma 3.4. ([2]) If K is a compact space and Y is a D-space, then K × Y is a D-space. Lemma 3.5. Let X be a DC-like space and Y a D-space. If X × Y is a D-product, then it is a D-space. Proof. Considering K1 = DC and K2 = D, it follows from Theorem 3.1 that X × Y is D(DC × D)-like. By Lemma 3.4, we have C × D ⊂ D. So we should notice that D(DC × D) = D(D(C × D)) ⊂ D(DD) = DD = D. Hence X × Y is D-like. By Lemma 3.3, X × Y is a D-space.
2
The following is an immediate consequence of Proposition 3.2 and Lemma 3.5. Corollary 3.6. If X is a subparacompact DC-like space and Y is a subparacompact D-space, then X × Y is a D-space.
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4. Hereditarily submetacompact C-scattered spaces Recall that a space X is submetacompact if for every open cover U of X, there is a sequence {Vn }n∈ω of open refinements of U, satisfying that for each x ∈ X, one can find nx ∈ ω such that Vnx is point-finite at x. Moreover, recall that a cover A of a space X is directed if for any A0 , A1 ∈ A, there is A2 ∈ A with A0 ∪ A1 ⊂ A2 . Using this, Junnila proved the following useful characterization of submetacompactness. Theorem 4.1. ([10, Theorem 4.4]) A space X is submetacompact if and only if every directed open cover of X has a σ-closure-preserving closed refinement. Recall that a space X is C-scattered if every non-empty closed subset F in X has a point with a compact neighborhood in F . For each F ∈ 2X \ {∅}, let F ∗ = {x ∈ F : x has no compact neighborhood in F }. Let X (0) = X and X (α+1) = (X (α) )∗ for each ordinal α. For a limit ordinal λ, let X (λ) =
α<λ
X (α) .
Fact 4.2. ([19]) Let X be a C-scattered space. Then the following are true. (i) X (α) = ∅ for some ordinal α, (ii) X (α) is locally compact if X (α+1) = ∅, (iii) each open (or closed) set in X is C-scattered. It follows from [20, Theorem 9.7] that a subparacompact C-scattered space is DC-like. It is easily seen by Theorem 3.1 and Proposition 3.2 that the subparacompact DC-like property is finitely productive. So Corollary 3.6 is a generalization of [11, Theorem 2.1] and [14, Corollaries 20 and 21]. The following seems to be known. However, for making sure, we put a simple proof. Lemma 4.3 (Folklore). If a space X is locally compact and submetacompact, then it has a σ-closure-preserving cover by compact sets. Proof. Let U be the family of all open sets U in X such that U is compact, then it is a directed open cover of X. By applying Theorem 4.1, we obtain a σ-closure preserving closed refinement F of U. Obviously, F is a cover of X by compact sets. 2 We also use the following, which was generalized in [9, Proposition 5.7]. Lemma 4.4. (Peng [12]) If a space X has a σ-closure-preserving closed cover by D-subspaces, then it is a D-space. Recall that a space X is hereditarily submetacompact if each subspace of X is submetacompact. Proposition 4.5. If X is a hereditarily submetacompact, C-scattered space and Y is a D-space, then X × Y is a D-space. Proof. By Fact 4.2(i), X (α) = ∅ for some ordinal α. We show by induction with respect to α. When α = 0, this is obvious because X = X (0) = ∅. Take an ordinal α, and assume that the proposition is true if X (β) = ∅ for β < α.
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Case 1. Let α be a successor. Let α = δ + 1. Since X (δ) is locally compact and submetacompact by Fact 4.2(ii), it follows from Lemma 4.3 that X (δ) has a σ-closure-preserving cover K by compact sets. Then {K × Y : K ∈ K} is a σ-closure-preserving closed cover of X (δ) × Y by D-subspaces. By Lemma 4.4, X (δ) × Y is a D-space. Take any closed subset E in X × Y disjoint from X (δ) × Y . By Fact 4.2(iii), X \ X (δ) is hereditarily submetacompact and C-scattered with (X \ X (δ) )(δ) = ∅. So it follows from the inductive assumption that (X \ X (δ) ) × Y is a D-space. Since E is a closed subset in (X \ X (δ) ) × Y , E is also a D-space. Hence X × Y is D-like. By Lemma 3.3, X × Y is a D-space. Case 2. Let α be a limit ordinal. Let U = {X \ X (β) : β < α}. Since U is directed, it follows from Theorem 4.1 that U has a σ-closure-preserving closed refinement F. Then note that each F ∈ F is hereditarily submetacompact and C-scattered such that F (βF ) = ∅ for some βF < α. By the inductive assumption, each F × Y is a D-space. Since {F × Y : F ∈ F} is a σ-closure-preserving closed cover of X × Y by D-subspaces, it follows from Lemma 4.4 that X × Y is a D-space. 2 5. Paracompact subspaces of an ordinal The following seems to be known. However, we state its proof for the reader’s convenience. Lemma 5.1 (Folklore). Let A be a subspace of an ordinal. Then the following are equivalent. (a) A is paracompact. (b) A is DC-like. (c) There is not an ordinal μ ∈ / A with cf μ > ω such that A ∩ μ is stationary in μ. / A with cf μ > ω such that A ∩ μ is stationary in μ. Since A ∩ μ Proof. (a) → (c): Assume that there is a μ ∈ is clopen in A, it is paracompact. Let U = {A ∩ [0, ξ] : ξ ∈ A ∩ μ}. Then U is an open cover of A ∩ μ. By the standard way using the Pressing Down Lemma, we see that U has no point-countable open refinement. This is a contradiction. (c) → (b): Assume that A is not DC-like. Take the smallest μ such that A ∩ [0, μ] is not DC-like. It is easy to see that μ ∈ / A. By the assumption, cf(μ) ≤ ω or A ∩ μ is not stationary in μ. In any case, we can represent A ∩ μ as ν∈cf(μ) Bν such that Bν ⊂ A ∩ [0, ξν ] for some ξν < μ. By the minimality of μ, each Bν is DC-like. Hence A ∩ μ = A ∩ [0, μ] is DC-like. This is a contradiction. (b) → (a): Since A is collectionwise normal, this immediately follows from [20, Theorem 10.6]. 2 Using the results above, we show the following. Proposition 5.2. If A is a paracompact subspace of an ordinal and Y is a submetacompact D-space, then A × Y is a D-space. Proof. Let A be a paracompact subspace of an ordinal μ ˆ, and Y a submetacompact D-space. For each μ≤μ ˆ, put Aμ = A ∩ [0, μ]. And put A−1 = ∅. Obviously, A−1 × Y = ∅ is a D-space and Aμˆ = A holds. By induction on an ordinal μ ≤ μ ˆ, we will prove that Aμ × Y is a D-space. After finishing induction, we conclude that A × Y = Aμˆ × Y is a D-space. Let μ be an ordinal and assume that Aν × Y is a D-space for each ν < μ. We show that Aμ × Y is a D-space. Case 1. μ ∈ / A. Since A is paracompact, we have that cf μ ≤ ω or Aμ = A ∩ μ is non-stationary in μ with cf μ > ω. Hence Aμ = j∈J Bj for some collection {Bj : j ∈ J} of subspaces of Aμ such that for each j ∈ J, there is
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a νj < μ with Bj ⊂ Aνj . Then Aνj × Y is a D-space by the inductive hypothesis, so Bj × Y is a D-space since it is a closed subspace of Aνj × Y , hence Aμ × Y = j∈J (Bj × Y ) is a D-space. Case 2. μ ∈ A. Let ϕ be an open neighborhood assignment for Aμ × Y . It suffices to find a closed discrete subset D in Aμ × Y such that Aμ × Y = {ϕ(d) : d ∈ D}. Put Z0 = {μ} × Y . Then Z0 is a closed set of Aμ × Y which is homeomorphic to Y . Since Y is a D-space, we can take a closed discrete subset D0 in Z0 (also in Aμ × Y ) such that D0 ⊂ Z0 ⊂ W0 , where let W0 = {ϕ(d) : d ∈ D0 }. Now, let Q = {Q ⊂ Y : Q is open in Y such that V × Q ⊂ W0 for some open neighborhood V of μ in Aμ }. Since Q is a directed open cover of Y which is submetacompact, it follows from Theorem 4.1 that Q has a σ-closure preserving closed refinement F. Enumerating members of F in natural, we have F = {Fξ : ξ < λ} for some ordinal λ and F<ξ := ζ<ξ Fζ is closed in Y for each ξ ≤ λ. Put Zξ = Z0 ∪ (Aμ × F<ξ ) for each ξ with 0 < ξ ≤ λ. We have Zλ = Aμ × Y since F<λ = F = Y . And {Zξ : ξ ≤ λ} is an increasing closed cover of Aμ × Y such that Zξ = ζ<ξ Zζ for each limit ordinal ξ ≤ λ. For each ξ < λ, there are a Qξ ∈ Q with Fξ ⊂ Qξ , an open neighborhood Vξ of μ in Aμ with Vξ × Qξ ⊂ W0 , and a νξ with −1 ≤ νξ < μ such that A ∩ (νξ , μ] ⊂ Vξ . Then (∗) Zξ+1 \ W is a D-space for each open set W in Aμ × Y with Zξ ∪ W0 ⊂ W . In fact, we should notice that Zξ+1 \ W ⊂ (Zξ+1 \ Zξ ) \ W0 ⊂ (Aμ × (F<(ξ+1) \ F<ξ )) \ (Vξ × Qξ ) ⊂ (Aμ × Fξ ) \ ((A ∩ (νξ , μ]) × Fξ ) = (Aμ \ (νξ , μ]) × Fξ ⊂ Aνξ × Y. By inductive hypothesis, Aνξ × Y is a D-space. Since Zξ+1 \ W is closed in Aμ × Y , it is also closed in Aνξ × Y . Hence (∗) is true. By induction on ξ ≤ λ, we will define an increasing sequence {Dξ : ξ ≤ λ} of closed discrete subsets in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ holds for each ξ ≤ λ, where Wξ = {ϕ(d) : d ∈ Dξ }, and Wζ ∩ Dξ ⊂ Dζ holds for each ζ, ξ with ζ < ξ ≤ λ. After finishing induction, we obtain a required closed and discrete set D in Aμ × Y by putting D = Dλ since Aμ × Y = Zλ ⊂ Wλ = {ϕ(d) : d ∈ Dλ } ⊂ Aμ × Y , thus we have Aμ × Y = {ϕ(d) : d ∈ Dλ } = {ϕ(d) : d ∈ D}. Let ξ ≤ λ and assume that an increasing sequence {Dζ : ζ ∈ ξ} of closed discrete subsets in Aμ × Y is defined, Dζ ⊂ Zζ ⊂ Wζ holds for each ζ ∈ ξ, and Wη ∩ Dζ ⊂ Dη holds for each η, ζ with η < ζ < ξ. We will find a closed discrete subset Dξ in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ holds, and Dζ ⊂ Dξ and Wζ ∩ Dξ ⊂ Dζ hold for each ζ < ξ. Case 2-1. ξ = 0. Dξ = D0 is already defined and satisfies the required condition. Case 2-2. ξ = ζ + 1 for some ζ < λ. It suffices to take a closed discrete subset Dξ in Aμ × Y such that Dξ ⊂ Zξ ⊂ Wξ , Dζ ⊂ Dξ and Wζ ∩ Dξ ⊂ Dζ hold. If such Dξ is taken, then it also holds for each η < ζ that Dη ⊂ Dξ and Wη ∩ Dξ ⊂ Dη since Dη ⊂ Dζ ⊂ Dξ and Wη ∩ Dξ = Wη ∩ Wζ ∩ Dξ ⊂ Wη ∩ Dζ ⊂ Dη hold by inductive hypothesis.
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A closed discrete set Dζ in Aμ × Y is already defined and D0 ⊂ Dζ and Zζ ⊂ Wζ hold, so Wζ = {ϕ(d) : d ∈ Dζ } is an open set in Aμ × Y with Zζ ∪ W0 ⊂ Wζ . By the above (∗), Zζ+1 \ Wζ is a D-space. Take a closed discrete set Eζ in Zζ+1 \ Wζ (also in Aμ × Y ) such that Zζ+1 \ Wζ ⊂ {ϕ(e) : e ∈ Eζ }. Define Dξ by putting Dξ = Dζ+1 = Dζ ∪ Eζ . Then Dξ is a closed discrete set in Aμ × Y since so are Dζ and Eζ . Obviously Dζ ⊂ Dξ holds, and we have Wζ ∩ Dξ ⊂ Dζ since Wζ ∩ Eζ = ∅. By Dζ ⊂ Zζ ⊂ Zξ , Eζ ⊂ Zζ+1 = Zξ , Wζ = {ϕ(d) : d ∈ Dζ } ⊂ {ϕ(d) : d ∈ Dξ } = Wξ and Zξ \ Wζ = Zζ+1 \ Wζ ⊂ {ϕ(e) : e ∈ Eζ } ⊂ {ϕ(e) : e ∈ Dξ } = Wξ , it follows that Dξ ⊂ Zξ ⊂ Wξ . Case 2-3. ξ is a limit ordinal. Put Dξ = ζ<ξ Dζ . By Zξ = ζ<ξ Zζ and Wξ = {ϕ(d) : d ∈ Dξ } = ζ<ξ ( {ϕ(d) : d ∈ Dζ }) = ζ<ξ Wζ , it follows from the inductive hypothesis that Dζ ⊂ Zζ ⊂ Zξ and Zζ ⊂ Wζ ⊂ Wξ hold for each ζ < ξ, and Wζ ∩ Dη ⊂ Wζ ∩ Dmax{ζ,η}+1 ⊂ Dζ for each ζ, η < ξ. Therefore Dξ ⊂ Zξ ⊂ Wξ holds, and Wζ ∩ Dξ ⊂ Dζ holds for each ζ < ξ. To see that Dξ is closed discrete in Aμ × Y , let z ∈ Aμ × Y . If z ∈ Wξ , then take a ζ ∈ ξ with z ∈ Wζ . Since Dζ is closed and discrete, there is an open neighborhood U of z with U ⊂ Wζ and |U ∩ Dζ | ≤ 1. By U ∩ Dξ ⊂ U ∩ Wζ ∩ Dξ ⊂ U ∩ Dζ , we have |U ∩ Dξ | ≤ 1. If z ∈ / Wξ , then (Aμ × Y ) \ Zξ is an open neighborhood of z which is disjoint from Dξ since Dξ ⊂ Zξ ⊂ Wξ and Zξ is a closed set in Aμ × Y . Hence Dξ is a closed discrete set in Aμ × Y . 2
Recall that every subspace of an ordinal is a GO-space. Douwen and Lutzer proved the following remarkable result. Theorem 5.3. ([4, Theorem 1.2]) A GO-space X is paracompact if and only if it is a D-space. By Proposition 5.2 and Theorem 5.3, we immediately have Corollary 5.4. Let A be a non-empty subspace of an ordinal, and Y a non-empty submetacompact space. Then A × Y is a D-space if and only if both A and Y are D-spaces. Remark 5.5. Each subspace of an ordinal is scattered, in particular, C-scattered. However, there is a paracompact subspace A of ω1 + 1 which is neither compact nor hereditarily submetacompact. In fact, let A = (ω1 + 1) \ {ω}. Since A satisfies Lemma 5.1(c), it is paracompact. Let B = ω1 \ {ω}. Then B is stationary in ω1 . By Lemma 5.1(c), B is not paracompact. Since paracompactness and submetacompactness are equivalent in the class of GO-spaces, the subspace B in A is not submetacompact. So Proposition 5.2 cannot be derived from Lemma 3.4, Corollary 3.6 or Proposition 4.5. 6. Countable products of subparacompact DC-like spaces Let m ∈ ω. A subset of the form i≤m Fi in a finite product i≤m Xi of spaces Xi , i ≤ m, is a closed rectangle if Fi is closed in Xi for each i ≤ m. Let X = i∈ω Xi be a countable product of spaces Xi , i ∈ ω. A subset of the form F = i∈ω Fi in it is called a closed (compact) box in X if Fi is a closed (compact) set in Xi for each i ∈ ω. Let and ≺ be binary relations on a set P . In particular, recall that is a partial order on P if • p p for each p ∈ P , • for each p, q, r ∈ P , if p q and q r, then p r.
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We call {pk : k ∈ ω} ⊂ P a -increasing sequence if pk pk+1 for each k ∈ ω, and we call {pk : k ∈ ω} ⊂ P a -decreasing sequence (≺-decreasing sequence) if pk+1 pk (pk+1 ≺ pk ) for each k ∈ ω. If is a partial order on P , then it is obvious that {pk : k ∈ ω} ⊂ P is a -increasing ( -decreasing) sequence if and only if pj pk (pk pj ) for every j ≤ k < ω. For Q ⊂ P , we call q a ≺-minimal element of Q if q ∈ Q and there is no p ∈ Q with p ≺ q. The following is our main result proved by using an elementary submodel. Theorem 6.1. If Xi is a subparacompact DC-like space for each i ∈ ω, then
i∈ω
Xi is a D-space.
Proof. Let X = i∈ω Xi , and let D X be the family of all closed discrete sets in X. Let ϕ be an open neighborhood assignment for X. Put ϕ[D] = {ϕ(d) : d ∈ D} for each D ⊂ X. To see that X is a D-space, it suffices to find a D ∈ D X with X = ϕ[D]. For D, D ∈ D X , let D < D denote that D ⊂ D and D ∩ ϕ[D] ⊂ D. Then < is a partial order on D X . Let Ψ denote the set of all functions ψ : D X → D X such that D < ψ(D) for every D ∈ D X . Let θ be a sufficiently large regular cardinal, and M a countable elementary submodel of Hθ such that X, τ (X), ϕ ∈ M, where recall that τ (X) is the topology on X. Let M ∩ Ψ = {ψk : k ∈ ω} and |{k ∈ ω : ψk = ψ}| = ω for every ψ ∈ M ∩ Ψ. By induction on k ∈ ω, we define a <-increasing sequence {Dk : k ∈ ω} ⊂ M ∩ D X by putting D0 = ∅ and Dk+1 = ψk (Dk ) for each k ∈ ω. After finishing induction, put Dω = k∈ω Dk . We will show that Dω ∈ D X and X = ϕ[Dω ]. The former is implied by the latter. Actually, for each x ∈ X, we can take j ∈ ω with x ∈ ϕ[Dj ]. Since Dj ∈ D X , there is an open neighborhood V of x such that V ⊂ ϕ[Dj ] and |V ∩Dj | ≤ 1. For each k with j < k < ω, since Dj < Dk , we have V ∩Dk ⊂ ϕ[Dj ] ∩Dk ⊂ Dj , so V ∩ Dω = j
Since M is an elementary submodel, each L ∈ M ∩ σL is expressed as L = n ∈ ω} ⊂ M ∩ L, so we have (M ∩ σL) = (M ∩ L). Put 2 F⊂ [x, M] = {E, F ∈
(M ∩ L) : x ∈ E} = {E, F ∈
n∈ω
Ln for some {Ln :
(M ∩ σL) : x ∈ E}.
2 2 Since X, X ∈ F⊂ and {X, X } ∈ M ∩ L, we have X, X ∈ F⊂ [x, M].
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2 Define a partial order ≤2 and a binary relation <2i , for each i ∈ ω, on F⊂ by
• E , F ≤2 E, F if and only if E ⊂ E and F ≤s F , • E , F <2i E, F if and only if E ⊂ E and F
˙ F˙ ∈ F 2 [x, M] with F˙ ≤s X and i ∈ ω. Then there is a <2 -minimal element E, ¨ F¨ of Claim 2. Let E, ⊂ i 2 2 ¨ ˙ ¨ ¨ ˙ ¨ F⊂ [x, M] such that E, F ≤ E, F and Ei ∩ si (Fi ) is compact. 2 Proof. By Claim 1, there is not a <2i -decreasing sequence {E (j) , F (j) : j ∈ ω} ⊂ F⊂ [x, M] starting from (0) (0) 2 2 ˙ F˙ . So there is a < -minimal element E , F of F⊂ [x, M] with E , F ≤2 E, ˙ F˙ . E , F = E, i ¨ Take L ∈ M ∩ L with E , F ∈ L . Then there is an L satisfying:
2 (··) L¨ ∈ σL and L¨ ⊂ {E , F ∈ F⊂ : Ei ∩ si (Fi ) is compact} hold. And for each E, F ∈ L , E ⊂ {E : ¨ E , F ≤2 E, F } holds. E , F ∈ L, To find an L¨ satisfying (··), let E, F be an arbitrary member of L . Since Xi is assumed to be subparacompact and si (Fi ) is the union of a discrete collection of compact subsets in Xi , we see that Ei is covered by a σ-discrete subcollection C of {C ⊂ Ei : C is a non-empty closed set in Xi such that C ∩ si (Fi ) is compact}. ¨ ¨ F¨ such that E ¨ is obtained by replacing the i-factor Ei of E with some Let L(E, F ) be the family of all E, ¨ C ∈ C, and F¨ = F . By putting L¨ = {L(E, F ) : E, F ∈ L }, we obtain an L¨ satisfying (··). Since the condition (··) can be expressed by a formula with parameters X, τ, s, i, L ∈ M, we can take ¨ Since E , F ∈ L and x ∈ E , L¨ ∈ M satisfying (··) (though it may be different from the old L). ¨ F¨ ∈ L¨ with E, ¨ F¨ ≤2 E , F ≤2 E, ˙ F˙ and x ∈ E. ¨ Since L¨ ∈ M ∩ σL, we have we obtain E, 2 2 2 ¨ ¨ F¨ is also a ¨ E, F ∈ F⊂ [x, M]. Since E , F is a
i≤m
Xi with the projection π≤m : X → X≤m .
2 Claim 3. There are an E ∗ , F ∗ ∈ F⊂ [x, M], an m ∈ ω, and a finite subfamily L∗ of
2 {E ∗∗ , F ∗∗ ∈ F⊂ : E ∗∗ , F ∗∗ <2i E ∗ , F ∗ }
i≤m
such that 2 (i) E ∗ , F ∗ is a <2i -minimal element of F⊂ [x, M] for each i ≤ m, (ii) for each k ∈ ω with m ≤ k, there is a Dk∗ ∈ D X such that Dk∗ ⊂ Dk ∪ E ∗ , Dk < Dk∗ and E ∗ ⊂ ϕ[Dk∗ ] ∪ E[L∗ ] . 2 Proof. By Claim 2, we can take a ≤2 -decreasing sequence {E (i) , F (i) : i ∈ {−1} ∪ ω} ⊂ F⊂ [x, M] starting (−1) (−1) (i) (i) 2 from E ,F = X, X such that for each i ∈ ω the pair E , F is a
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• Vy ⊂ ϕ(y), • if y ∈ ϕ[Dω ], then Vy ⊂ ϕ[Dky ] for some ky ∈ ω, • if y ∈ / i∈ω E (i) , then Vy ∩ E (iy ) = ∅ for some iy ∈ ω. −1 We may assume that Vy = π≤m (Uy ) for an my ∈ ω and an open set Uy in X≤my . Since K is compact, y there is a finite subset J of K such that K ⊂ y∈J Vy . Take an m ∈ ω such that my ≤ m for each y ∈ J, ky ≤ m for each y ∈ J ∩ ϕ[Dω ], iy ≤ m for each y ∈ J \ i∈ω E (i) , and if Kj = ∅ for some j ∈ ω, then j ≤ m for the least such j. We show that E ∗ , F ∗ = E (m) , F (m) satisfies the required condition. For each i ≤ m, since E (i) , F (i) 2 [x, M] and E ∗ , F ∗ ≤2 E (i) , F (i) , we see that F ∗ i = F (i) i and E ∗ , F ∗ is is a <2i -minimal element of F⊂ ∗ also a <2i -minimal element of F⊂ [x, M]. Put
U=
{u ∈ X≤m : u {0, · · · , my } ∈ Uy },
y∈J −1 and put V = π≤m (U ). Then U and V are open sets in X≤m and X, respectively, such that K≤m := Vy . So X≤m has a finite cover by closed rectangles which are contained in U i≤m Ki ⊂ U and V = y∈J or disjoint from K≤m . And if i≤m Ei is a closed rectangle in X≤m which is disjoint from K≤m , then there is an i0 ≤ m such that Ei0 ∩Ki0 = ∅, so Ei0 ∩E ∗ i0 ⊂ Ei0 ∩E (i0 ) i0 ⊂ F (i0 ) i0 \si0 (F (i0 ) i0 ) = F ∗ i0 \si0 (F ∗ i0 ). If E ∗∗ ∈ F is obtained from E ∗ by replacing i-factor E ∗ i with Ei ∩E ∗ i for each i ≤ m, and F ∗∗ ∈ F is obtained from F ∗ by replacing the i0 -factor Fi∗0 with Ei0 ∩ Ei∗0 , then E ∗∗ , F ∗∗ <2i0 E ∗ , F ∗ holds. Hence there is a 2 finite subfamily L∗ of i≤m {E ∗∗ , F ∗∗ ∈ F⊂ : E ∗∗ , F ∗∗ <2i E ∗ , F ∗ } such that E ∗ ⊂ V ∪ E[L∗ ] . Put D∗ = J ∩( i∈ω E (i) ) \ϕ[Dω ]. Put Dk∗ = Dk ∪D∗ for each k ∈ ω with m ≤ k. We have Dk∗ ∈ D X since D∗ is a finite set. By D∗ ⊂ i∈ω E (i) ⊂ E (m) = E ∗ , we have Dk∗ ⊂ Dk ∪E ∗ . By ϕ[Dk ] ∩D∗ ⊂ ϕ[Dω ] ∩D∗ = ∅, we have ϕ[Dk ] ∩ Dk∗ ⊂ Dk , therefore Dk < Dk∗ holds. Let y ∈ J. If y ∈ i∈ω E (i) and y ∈ / ϕ[Dω ], then y ∈ D∗ ⊂ Dk∗ , so Vy ∩ E ∗ ⊂ Vy ⊂ ϕ(y) ⊂ ϕ[Dk∗ ]. If y ∈ ϕ[Dω ], then since ky ≤ m ≤ k, we have Dky ⊂ Dk ⊂ Dk∗ , so Vy ∩ E ∗ ⊂ Vy ⊂ ϕ[Dky ] ⊂ ϕ[Dk∗ ]. If y ∈ / i∈ω E (i) , then since iy ≤ m, we have E ∗ = E (m) ⊂ E (iy ) , so Vy ∩ E ∗ ⊂ Vy ∩ E (iy ) = ∅ ⊂ ϕ[Dk∗ ]. Hence, V ∩ E ∗ = y∈J (Vy ∩ E ∗ ) ⊂ ϕ[Dk∗ ]. By E ∗ ⊂ V ∪ E[L∗ ], we have E ∗ ⊂ ϕ[Dk∗ ] ∪ E[L∗ ] . 2
Take E ∗ , F ∗ , m and L∗ described in Claim 3. And take an L ∈ M ∩ L such that E ∗ , F ∗ ∈ L. Claim 4. There is a ψ such that: (∗) ψ ∈ Ψ holds. And for each D ∈ D X and D = ψ(D), there is an L such that: (∗∗) L ∈ L holds. And for each E, F ∈ L, if (†) there are a D† ∈ D X and a finite subfamily L† of
2 {E † , F † ∈ F⊂ : E † , F † <2i E, F }
i≤m
such that D† ⊂ D ∪ E, D < D† and E ⊂ ϕ[D† ] ∪ E[L† ] , then E ⊂ ϕ[D ] ∪ i≤m {E : E , F ∈ L , E , F <2i E, F }. Proof. It suffices to show that for each D ∈ D X , there are a D ∈ D X with D < D and an L satisfying the condition (∗∗). Let L0 be the family of all E, F ∈ L satisfying the condition (†). If L0 = ∅, then D = D and L = ∅ satisfy the required condition. So assume that L0 = ∅. For each E, F ∈ L0 , fix a † † † † D(E,F ) , L(E,F ) = D , L witnessing (†). Put
Y. Hirata, Y. Yajima / Topology and its Applications 195 (2015) 297–311
D =
† {D(E,F ) : E, F ∈ L0 } and L =
307
{L†(E,F ) : E, F ∈ L0 }.
† † Let E, F ∈ L0 . Then D(E,F ) \D ⊂ E and ϕ[D] ∩D(E,F ) ⊂ D. By L ∈ L, E[L] = {E : E, F ∈ L} is locally † finite in X, so we have D ∈ D X and L ∈ L. And ϕ[D] ∩D = {ϕ[D] ∩D(E,F ) : E, F ∈ L0 } ⊂ D holds, so † † we have D < D . By the definition, it is obvious that E ⊂ ϕ[D(E,F ) ] ∪ E[L(E,F ) ] ⊂ ϕ[D ] ∪ i≤m {E : E , F ∈ L , E , F <2i E, F } holds for each E, F ∈ L satisfying (†). 2
Since the condition (∗) in Claim 4 can be expressed by a formula with parameters X, τ, s, m, ϕ, L ∈ M, we can take ψ ∈ M satisfying (∗) (though it may be different from the old ψ). Take a k ∈ ω with ψk = ψ and m ≤ k. Put D = Dk and D = Dk+1 = ψk [Dk ] = ψ[D]. By the definition of ψ, there is an L satisfying the condition (∗∗) in Claim 4. Since the condition (∗∗) can be expressed by a formula with parameters X, τ, s, m, ϕ, L, D, D ∈ M, we can take L ∈ M satisfying (∗∗). Then D† = Dk∗ and L† = L∗ in Claim 3 2 witness that (†) holds for E, F = E ∗ , F ∗ . It follows from E ∗ , F ∗ ∈ F⊂ [x, M] that x ∈ E ∗ ⊂ ϕ[D ] ∪
{E : E , F ∈ L , E , F <2i E ∗ , F ∗ }.
i≤m
2 By L ∈ M ∩L, we have E , F ∈ (M ∩L) for each E , F ∈ L . On the other hand, E , F ∈ / F⊂ [x, M] 2 ∗ ∗ ∗ ∗ 2 2 if E , F
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cs,t ∈ D(s, t) with cs,t = ds,t and γ(s, t) ∈ ω1 with cs,t ∈ Fγ(s,t) . If D(s, t) = ∅, then put α(s, t) = 0. If |D(s, t)| = 1, then put α(s, t) = δ(s, t). And if |D(s, t)| ≥ 2, then put α(s, t) = max{δ(s, t), γ(s, t)}. Inductively, we take a strictly increasing sequence {βn : n ∈ ω} ⊂ ω1 . For n = 0, put β0 = 0. Let n ∈ ω and assume that βn ∈ ω1 is defined. Take βn+1 ∈ ω1 such that βn < βn+1 and α(s, t) < βn+1 for each finite subset s of βn and for each t ∈ N s . After finishing induction, let β = sup{βn : n ∈ ω}. Take y ∈ Fβ and d ∈ D with y ∈ V (d). Take the α ∈ ω1 with d ∈ Fα . We have y(α) = 0 by y ∈ V (d), so β ≤ α holds by y ∈ Fβ . Hence there is a z ∈ Fβ with z β = d β. Let W be an arbitrary open neighborhood of z in F . Then there is a finite subset sˆ of ω1 such that {x ∈ F : x sˆ = z sˆ} ⊂ W . Put s = sˆ ∩ β and t = d s. Then D(s, t) is non-empty since d ∈ D(s, t). So ds,t ∈ D(s, t) and δ(s, t) ∈ ω1 with ds,t ∈ Fδ(s,t) are defined. Take an m ∈ ω with s ⊂ βm . We have δ(s, t) ≤ α(s, t) < βm+1 < β ≤ α. In particular, δ(s, t) = α, so Fδ(s,t) ∩ Fα = ∅, thus ds,t = d. Hence, |D(s, t)| ≥ 2. So cs,t ∈ D(s, t) with cs,t = ds,t and γ(s, t) ∈ ω1 with c(s, t) ∈ Fγ(s,t) are defined. And we have γ(s, t) ≤ α(s, t) < βm+1 < β. Since ds,t , cs,t ∈ D(s, t) and s ⊂ β, we have ds,t s = cs,t s = t = d s = z s. For each ξ ∈ sˆ \ s, since δ(s, t), γ(s, t) < β ≤ ξ, ds,t ∈ Fδ(s,t) , cs,t ∈ Fγ(s,t) and z ∈ Fβ , we have ds,t (ξ) = cs,t (ξ) = 0 = z(ξ). Therefore ds,t sˆ = cs,t sˆ = z sˆ holds, thus ds,t , cs,t ∈ D ∩ W . We see that D ∩ W has at least two members for each open neighborhood W of z, hence D is not a closed discrete subset in F . Next, we will show that F is dually discrete (this case seems to be more difficult than the former). Let ϕ be an open neighborhood assignment for F . We would like to find a discrete subset D in F with {ϕ(d) : d ∈ D} = F . For each x ∈ F , take and fix a finite subset sx of ω1 such that {y ∈ F : y sx = x sx } ⊂ ϕ(x). Let δ ≤ ω1 . Set S(δ) = {s, t : s is a finite subset of δ, t ∈ N s }. For each s, t ∈ S(ω1 ), put Fα (s, t) = {x ∈ Fα : sx ∩α = s, x s = t} and A(s, t) = {α ∈ ω1 : Fα (s, t) = ∅}. And set S0 (δ) = {s, t ∈ S(δ) : A(s, t) is bounded in ω1 }, S1 (δ) = {s, t ∈ S(δ) : A(s, t) is unbounded in ω1 }. We denote S(ω1 ), S0 (ω1 ) and S1 (ω1 ) by S, S0 and S1 , respectively. Take a ξ ∈ ω1 . For each s, t ∈ S(ξ) = S0 (ξ) ∪ S1 (ξ), take a βξ (s, t) ∈ ω1 such that A(s, t) ⊂ βξ (s, t) and ξ < βξ (s, t) if s, t ∈ S0 (ξ) and that βξ (s, t) ∈ A(s, t) ∩ (ξ, ω1 ) if s, t ∈ S1 (ξ). Moreover, take a βξ ∈ ω1 with βξ (s, t) < βξ for each s, t ∈ S(ξ). Put C = {δ ∈ ω1 : δ is a limit ordinal such that ξ < δ implies βξ < δ}. Then C is a closed unbounded set in ω1 . It is easily verified that for each δ ∈ C, • A(s, t) ⊂ δ holds for each s, t ∈ S0 (δ), • A(s, t) ∩ (γ, δ) = ∅ holds for each s, t ∈ S1 (δ) and each γ with −1 ≤ γ < δ. And take a pairwise disjoint family {C(s, t) : s, t ∈ S1 } of unbounded subsets in ω1 such that C(s, t) ⊂ {γ ∈ C : s ⊂ γ} for each s, t ∈ S1 . Let γ ∈ C ∪ {−1}. Put δγ = min{δ ∈ C : γ < δ} and F ∗ [γ] = γ<α<δγ Fα . By enumerating the elements, let S(δγ ) = {sγ,n , tγ,n : n ∈ ω}. We inductively define an increasing sequence {Dγ,n : n ∈ ω} of finite subsets in F ∗ [γ]. For n = 0: If γ ∈ C(sγ , tγ ) for some sγ , tγ ∈ S1 , then by δγ ∈ C, γ < δγ and sγ ⊂ γ ⊂ δγ , we have sγ , tγ ∈ S1 (δγ ), and so we can take an αγ ∈ A(sγ , tγ ) with γ < αγ < δγ . By Fαγ (sγ , tγ ) = ∅, we obtain a
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309
dγ ∈ Fαγ (sγ , tγ ). Then dγ ∈ Fαγ ⊂ F ∗ [γ] holds. Define Dγ,0 by putting Dγ,0 = {dγ }. If γ ∈ / C(s, t) for any s, t ∈ S1 , then let Dγ,0 = ∅. For n + 1: Let n ∈ ω and assume that a finite subset Dγ,n of F ∗ [γ] is defined. If there is an x ∈ F ∗ [γ] \ {ϕ(d) : d ∈ Dγ,n } such that sx ∩ δγ = sγ,n and x sγ,n = tγ,n , then take and fix one of such dγ,n = x and let Dγ,n+1 = Dγ,n ∪ {dγ,n }. If there is no such x, then let Dγ,n+1 = Dγ,n . After finishing induction, let Dγ = n∈ω Dγ,n . By the definition, Dγ is a discrete subset in F ∗ [γ]. Let D = γ∈C∪{−1} Dγ . We show that D is discrete. Since Dγ is a discrete subset in F ∗ [γ] for each γ, it suffices to show that {F ∗ [γ] : γ ∈ C ∪ {−1}} is a discrete collection in F ∗ := γ∈C∪{−1} F ∗ [γ]. Let x ∈ F ∗ [γ] for some γ ∈ C ∪ {−1}. Then x ∈ Fα for some α with γ < α < δγ . By the definition of Fα , we have x(γ) = 0 and x(α) = 0. We obtain an open neighborhood V := {y ∈ F : y(γ) = 0, y(α) = 0} of x which is disjoint from F ∗ [γ ] for any γ ∈ C ∪ {−1} with γ = γ. Actually, let x ∈ F ∗ [γ ]. Then x ∈ Fα for some α with γ < α < δγ . If γ < γ , then α < δγ ≤ γ < α by the minimality of δγ , so we have x (α) = 0. If γ > γ , then γ ≥ δγ > α by the minimality of δγ , so we have x (γ) = 0. In any case, we obtain x ∈ / V. To see that F is dually discrete, it suffices to show that {ϕ(d) : d ∈ D} = F holds. Let x ∈ F . We would like to find a d ∈ D with x ∈ ϕ(d). Take the α ∈ ω1 with x ∈ Fα . And let s = sx ∩ α and t = x s. Then s, t ∈ S(α) = S0 (α) ∪ S1 (α). In case s, t ∈ S0 (α): By x ∈ Fα (s, t), we have α ∈ A(s, t) \ α, thus A(s, t) ⊂ α. Hence, α ∈ / C and so γ < α < δγ for some γ ∈ C ∪ {−1}, that is x ∈ F ∗ [γ]. Take an m ∈ ω with sγ,m = sx ∩ δγ and tγ,m = x sγ,m . If x ∈ {ϕ(d) : d ∈ Dγ,m }, then we obtain a d ∈ Dγ,m ⊂ Dγ ⊂ D with x ∈ ϕ(d), and such d is a required one. Otherwise, dγ,m ∈ Dγ,m+1 ⊂ Dγ ⊂ D is defined. Let d = dγ,m . By the definition, we have d ∈ F ∗ [γ], sd ∩ δγ = sγ,m and x sγ,m = tγ,m = d sγ,m . There is an α with γ < α < δγ and d ∈ Fα . For each ξ ∈ sd \ δγ , it follows from x ∈ Fα , d ∈ Fα and α, α < δγ ≤ ξ that x(ξ) = d(ξ) = 0. Therefore, x sd = d sd holds, thus x ∈ ϕ(d). Hence d is a required one. In case s, t ∈ S1 (α): Take a γ ∈ C(s, t) with α ≤ γ. Then sγ , tγ = s, t , an ordinal αγ with γ < αγ < δγ , and a dγ ∈ Fαγ (s, t) are defined. Let d = dγ . We have d ∈ Dγ,0 ⊂ Dγ ⊂ D. It follows from d ∈ Fαγ (s, t) that sd ∩ αγ = s and x s = t = d s. And for each ξ ∈ sd \ αγ , it follows from x ∈ Fα , d ∈ Fαγ and α ≤ γ < αγ ≤ ξ that x(ξ) = d(ξ) = 0. Therefore, x sd = d sd , thus x ∈ ϕ(d) holds. Hence d is a required one. 2 Since every closed subspace of a D-space is also a D-space, we immediately have Corollary 7.2. The space N ω1 is not a D-space. Moreover, combining this with Theorem 6.1, the following is obtained. Corollary 7.3. Let λ∈Λ Xλ be an uncountable product of non-empty subparacompact DC-like spaces. Then λ∈Λ Xλ is a D-space if and only if each Xλ is compact except countably many λ ∈ Λ. Proof. The “if” part follows from Lemma 3.4 and Theorem 6.1. Conversely, assume that there is Γ ⊂ Λ such that |Γ| = ω1 and Xλ is not compact for each λ ∈ Γ. Since each Xλ is subparacompact, such an Xλ is not countably compact. So there is an infinite countable closed discrete subset Nλ in Xλ for each λ ∈ Γ. Then to a closed subspace in λ∈Λ Xλ . It follows from Corollary 7.2 that λ∈Γ Nλ is λ∈Γ Nλ is homeomorphic not a D-space. Hence λ∈Λ Xλ is not a D-space. 2 Since each countably compact and submetacompact space is compact, the following is similarly obtained. Corollary 7.4. Let X be a submetacompact space and κ an uncountable cardinal. Then X κ is a D-space if and only if X is compact.
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8. Problems Since subparacompactness implies submetacompactness, it is natural from Corollary 3.6 to raise Problem 1. Let X be a DC-like space and Y a D-space. If X is subparacompact and Y is submetacompact, or if X is submetacompact and Y is subparacompact, is then X × Y a D-space? From the negative aspect of this, we raise Problem 2. Are there a DC-like space X and a D-space Y such that X × Y is not a D-space? Remark 8.1. Every binary open cover of a normal and rectangular product X × Y has a σ-locally finite refinement by closed rectangles. Using this property instead of D-products, we can show the following: Let X be a DC-like space and Y a D-space. If X × Y is normal and rectangular, then it is a D-space. Since subparacompact C-scattered spaces are DC-like, it is natural from Proposition 4.5 to ask Problem 3. Let X be a submetacompact C-scattered space and Y a submetacompact D-space. Is then X ×Y a D-space? Since every subspace of an ordinal is a GO-space, it is natural from Proposition 5.2 to ask Problem 4. If X and Y are paracompact GO-spaces, is then X × Y D-space? As well as Problem 1, it is natural from Theorem 6.1 to raise Problem 5. Let Xi be a submetacompact DC-like (or C-scattered) space for each i ∈ ω. Is then D-space?
i∈ω
Xi a
Remark 8.2. We cannot expect an analogical result to Theorem 3.1 for countable products. In fact, consider the irrational space N ω . It was shown in [6] that N ω is not totally paracompact (i.e., it has a base which has no locally finite subcover). Obviously, N is a 1-like space. Since it follows from [21, Theorem 5.1] that every paracompact DC-like space is totally paracompact. Hence N ω is not DC-like. In particular, N ω is not D1-like. Finally, it is natural to raise the following problem from the previous section. Problem 6. Is the space N ω1 dually discrete? References [1] O.T. Alas, L.R. Junqueira, G. Wilson, Dually discrete spaces, Topol. Appl. 155 (2008) 1420–1425. [2] C.R. Borges, A.C. Wehrly, A study of D-spaces, Topol. Proc. 16 (1991) 7–15. [3] J. Chaber, G. Gruenhage, R. Pol, On Souslin sets and embeddings in integer-valued function spaces on ω1 , Topol. Appl. 82 (1998) 71–104. [4] E.K. van Douwen, D.J. Lutzer, A note on paracompactness in generalized ordered spaces, Proc. Am. Math. Soc. 125 (1997) 1237–1245. [5] E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pac. J. Math. 81 (1979) 371–377. [6] B. Fitzpatrick Jr., R.M. Ford, On the equivalence of small and large inductive dimension in certain metric spaces, Duke Math. J. 34 (1967) 33–37. [7] F. Galvin, R. Telgársky, Stationary strategies in topological games, Topol. Appl. 22 (1986) 51–69. [8] G. Gruenhage, A survey of D-spaces, Contemp. Math. 533 (2011) 13–28. [9] H. Guo, H. Junnila, D-spaces and thick covers, Topol. Appl. 158 (2011) 2111–2121.
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