Separation of diagonal in monotonically normal spaces and their products

Topology and its Applications 196 (2015) 1033–1059

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Topology and its Applications www.elsevier.com/locate/topol

Separation of diagonal in monotonically normal spaces and their products ✩ Yasushi Hirata, Yukinobu Yajima ∗ Department of Mathematics, Kanagawa University, Yokohama, 221-8686, Japan

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Article history: Received 25 December 2013 Received in revised form 1 April 2014 Accepted 7 April 2014 Available online 8 June 2015 MSC: primary 54B10, 54D15, 54D20 secondary 03E10 Keywords: Δ-paracompact Δ-normal Normal Monotonically normal Orthocompact Countable tightness Countable extent The κ-dop property The S-docs property

a b s t r a c t As separation of diagonal, we study when monotone normality implies Δ-paracompactness or Δ-normality. For that, it is proved that every monotonically normal space is Δ-paracompact if the projection of its square is closed. Moreover, it is proved that every monotonically normal space is Δ-normal if it has countable tightness (or countable extent). In particular, the parenthetic part is an affirmative answer to Burke and Buzyakova’s problem in 2010. Secondly, we study the relation between normality and Δ-paracompactness or Δ-normality in certain products. For that, we additionally introduce two new neighborhood properties. Using these ones, it is proved that the product X × K of a monotonically normal space X and a compact space K is Δ-paracompact (respectively, Δ-normal) if and only if X is Δ-paracompact (respectively, Δ-normal) and X × K is normal. © 2015 Elsevier B.V. All rights reserved.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Monotone normality and related results . . . . . . . . . . 3. Δ-paracompactness on monotonically normal spaces . . 4. Δ-normality on monotonically normal spaces . . . . . . . 5. A neighborhood property for Δ-paracompactness . . . . 6. Another neighborhood property for Δ-normality . . . . 7. A characterization for Δ-paracompactness of products 8. A characterization for Δ-normality of products . . . . . DC-like spaces . . . . . . . . . . . . . . . . . . 9. Paracompact DC References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dedicated to the memory of Professor Ratislav Telgársky.

* Corresponding author. E-mail addresses: [email protected] (Y. Hirata), [email protected] (Y. Yajima). http://dx.doi.org/10.1016/j.topol.2015.05.027 0166-8641/© 2015 Elsevier B.V. All rights reserved.

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1. Introduction Throughout this paper, all spaces are assumed to be Hausdorff. For a space X, the diagonal {x, x : x ∈ X} of X is denoted by ΔX . Note that the diagonal ΔX of X is closed in X × X. Buzyakova [3] gave the unified names for several diagonal separation properties, and studied their implications. Here, we take up the two properties of Δ-paracompactness and Δ-normality from these properties, because they seem to be essential. Definition 1. A space X is Δ-paracompact if for each closed set C in X × X disjoint from ΔX , there is a  locally finite open cover U of X such that {U × U : U ∈ U} misses C. Definition 2. A space X is Δ-normal if for each closed set C in X × X disjoint from ΔX , there are disjoint open sets U and V in X × X such that C ⊂ U and ΔX ⊂ V . As pointed out in [3,13], in the class of normal spaces, Δ-paracompactness implies Δ-normality and coincides with functional Δ-paracompactness, which was known from long time ago and was called divisibility or strong collectionwise normality in other words (see [5]). Buzyakova [3] proved that every GO-space is Δ-paracompact. Immediately after, Burke and Buzyakova [2] proved that first countable, countably compact and monotonically normal space is Δ-paracompact. The concept of Δ-normality was named and studied by Hart [7], who showed by a simple example that there is a monotonically normal space which is not Δ-normal. As long as seeing these results, one may consider there is a big difference between monotone normality and these diagonal separation properties. However, to find a nice class of spaces for these diagonal separation properties, it is natural to consider the following problem. Problem A. When is a monotonically normal space Δ-paracompact or Δ-normal? In Section 2, as the preparation of proofs later, we state some notation and results for monotonically normal spaces. By Hart’s example stated above, Problem A is negative without any additional condition. In Section 3, answering Problem A for Δ-paracompactness, it is proved that every monotonically normal space is Δ-paracompact if the projection of its square is closed. In Section 4, answering Problem A for Δ-normality, it is proved that every monotonically normal space is Δ-normal if it has countable tightness (or countable extent). This parenthetic part is an affirmative answer to [2, Problem 2.2]. Hart [7] also showed that the non-normal product ω1 × (ω1 + 1) is not Δ-normal. On the other hand, normality of the products of a monotonically normal space and a compact space was characterized by some neighborhood properties in [9]. So we consider the following natural problems. Problem B. Let X be a Δ-paracompact and monotonically normal space and K a compact space. (1) If X × K is Δ-paracompact or Δ-normal, is it normal? (2) If X × K is normal, is it Δ-paracompact or Δ-normal? In Sections 5 and 6, we introduce two new neighborhood properties and give affirmative answers to Problem B(1) for Δ-paracompactness and Δ-normality, respectively. In Sections 7 and 8, we also give affirmative answers to Problem B(2) for Δ-paracompactness and Δ-normality, respectively. Consequently, we can obtain two characterizations for Δ-paracompactness and Δ-normality of such products in terms of normality. In

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Section 9, compactness is generalized as paracompact DC-likeness. We also characterize Δ-paracompactness and Δ-normality of the products with paracompact DC-like factor in terms of neighborhood properties. 2. Monotone normality and related results Definition 3. A space X is said to be monotonically normal if for any two disjoint closed sets E and F in X, one can assign an open set M (E, F ), satisfying that (i) E ⊂ M (E, F ) ⊂ ClX M (E, F ) ⊂ X \ F , (ii) if E  and F  are disjoint closed sets in X with E ⊂ E  and F ⊃ F  , then M (E, F ) ⊂ M (E  , F  ) holds. The function M is called a monotone normality operator for X. Lemma 2.1. ([10]) A space X is monotonically normal if and only if for each open set U in X and for each x ∈ U , one can assign an open set H(x, U ) in X, satisfying that (i) x ∈ H(x, U ) ⊂ U , (ii) H(x, U ) ∩ H(y, V ) = ∅ implies that x ∈ V or y ∈ U . For convenience, the function H is called a monotone normality assignment for X. These M and H are sometimes used in a couple of later proofs. Let X be a space. For each regular uncountable cardinal κ, we let S(X, κ) = {E ⊂ X : E is closed in X and homeomorphic to a stationary subset in κ}. Moreover, we let S ∗ (X) = {κ : κ is a regular uncountable cardinal with S(X, κ) = ∅},  S(X) = {S(X, κ) : κ ∈ S ∗ (X)}. For each E ∈ S(X, κ) with κ ∈ S ∗ (X), we assign a stationary subset SE in κ and a homeomorphism eE from SE onto E, and fix them. Then the assignments of SE and eE do not matter their choices (see [9]). The notation stated above will play very important roles for a monotonically normal space X, because of the following strong result. Theorem 2.2. (Balogh and Rudin [1]) Let X be a monotonically normal space. For every open cover U of X, there are a σ-disjoint partial refinement V of U by open sets in X and a discrete collection F of closed sets   belonging to S(X) such that X \ V = F. Making use of Theorem 2.2, the following lemma was proved. Lemma 2.3. ([17, Lemma 8.2]) Let X be a monotonically normal space and K a compact space. Let O be an open cover of X × K, satisfying that for each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each y ∈ K, there is an open rectangle P × Q in X × K such that eE (SE ∩ (γ, κ)) ⊂ P for some γ ∈ κ, y ∈ Q and P × Q is contained in some member of O. Then there is a locally finite open (cozero) cover U of X and a family {VU : U ∈ U} of finite open (cozero) covers of K such that {U × V : U ∈ U and V ∈ VU } refines O.

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Considering K as a one-point space, we immediately have Corollary 2.4. ([9, Corollary 2.7]) Let X be a monotonically normal space and U an open cover of X. Then U has a locally finite open refinement (that is, U is a normal cover of X) if and only if for each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X), there is a U ∈ U such that eE (SE ∩ (γ, κ)) ⊂ U for some γ ∈ κ. Conversely, Lemma 2.3 is easily derived by this corollary. 3. Δ-paracompactness on monotonically normal spaces Orthocompactness The following is so convenient that it will be often used in later proofs. Lemma 3.1. A space X is Δ-paracompact if and only if for every open cover G of X, there is a locally finite open cover U of X such that: for each x0 , x1 ∈ U with U ∈ U, there is G ∈ G with x0 , x1 ∈ G. As the proof is straightforward, it is left to the reader. The following is also useful to consider our Problem A for Δ-paracompactness. Lemma 3.2. For a monotonically normal space X, the following are equivalent: (a) X is Δ-paracompact.  (b) For each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each family G of open sets in X with E ⊂ G,   there is an open set U in X with eE SE ∩ (γ, κ) ⊂ U for some γ ∈ κ such that: for each x0 , x1 ∈ U , there is a G ∈ G with x0 , x1 ∈ G. (c) For each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each open cover G of X, there is an open set   U in X with eE SE ∩ (γ, κ) ⊂ U for some γ ∈ κ such that: for each x0 , x1 ∈ U , there is a G ∈ G with x0 , x1 ∈ G.  Proof. (a) → (b): Since X is normal, we can take an open set P in X with E ⊂ P ⊂ ClX P ⊂ G. By applying Δ-paracompactness for an open cover G ∪ {X \ ClX P } of X, we obtain a locally finite open cover U of X such that: for each x0 , x1 ∈ U  ∩ P with U  ∈ U, there is a G ∈ G with x0 , x1 ∈ G. Since U is   point-countable, there is a U0 ∈ U with eE SE ∩ (γ, κ) ⊂ U0 for some γ ∈ κ. Then U = U0 ∩ P satisfies the required condition. (b) → (c): Trivial. (c) → (a): Let G be an open cover of X. And let U be the family of all open sets U in X such that: for each x0 , x1 ∈ U with U ∈ U, there is a G ∈ G with x0 , x1 ∈ G. By G ⊂ U, U is also an open cover of X. By the assumption, it follows from Corollary 2.4 that U has a locally finite subcover (open refinement). By Lemma 3.1, X is Δ-paracompact. 2 

Recall that a space X is orthocompact if every open cover U of X has an open refinement W such that W  is open in X for each W  ⊂ W, where such a W is said to be interior-preserving.

Lemma 3.3. For a monotonically normal space X, the following are equivalent: (a) X is orthocompact. (b) For each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each family G of open sets in X with  E ⊂ G, there is an ascending sequence {Uξ : ξ ∈ κ} of open sets in X which partially refines G such    that eE SE ∩ (γ, κ) ⊂ ξ∈κ Uξ for some γ ∈ κ.

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(c) For each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each open cover G of X, there is an interior   preserving family U of open sets in X which partially refines G such that eE SE ∩ (γ, κ) ⊂ U for some γ ∈ κ. Proof. (a) → (b): Take an interior-preserving open refinement U of an open cover G ∪ {X \ E}. By PDL (= the Pressing Down Lemma), there is a γ ∈ κ such that {Eξ : ξ ∈ κ} partially refines U, where Eξ = ∅    for each ξ ≤ γ, and Eξ = eE SE ∩ (γ, ξ] for each ξ ∈ (γ, κ). Let Uξ = {U ∈ U : Eξ ⊂ U } for each ξ ∈ κ. Then {Uξ : ξ ∈ κ} satisfies the required condition. (b) → (c): An ascending sequence U = {Uξ : ξ ∈ κ} in (b) is an interior-preserving family of open sets, so it satisfies the condition (c). (c) → (a): Let G be an open cover of X. And let H be the family of all open sets H in X such that  H ⊂ UH for some interior preserving open partial refinement UH of G. Then H is an open cover of X. By Corollary 2.4, we can take a locally finite subcover (open refinement) H0 of H. Then {H ∩ U : H ∈ H0 and U ∈ UH } is an interior-preserving open refinement of G. Hence X is orthocompact. 2 Remark 3.4. Monotone normality of X in the assumptions of Lemmas 3.2 and 3.3 are used only for deriving (c) → (a). It suffices only assuming normality for Lemma 3.2(a) → (b). For the other implications, we do not need any assumption for a space X. Since Lemma 3.3(b) implies Lemma 3.2(b), we obtain the following result. Corollary 3.5. ([6,14]) Every monotonically normal and orthocompact space is Δ-paracompact. Remark 3.6. It follows from [6, Theorem 2.0] and Lemma 3.1 that Δ-paracompact and normal spaces are equivalently almost 2-fully normal. So Corollary 3.5 had been proved in [14, Theorem 6]. Closed projections of squares Here we give a generalization of Burke and Buzyakova’s result in [2]. Lemma 3.7. Let X and Y be spaces with A ⊂ X and B ⊂ Y , and let O be an open set in X × Y with A × B ⊂ O. (1) If the projection π0 : X × B → X is a closed map, then there is an open set P in X with A ⊂ P such that P × B ⊂ O. (2) If the projection π1 : A × Y → Y is a closed map, then there is an open set Q in Y with B ⊂ Q such that A × Q ⊂ O. Proof. (1) Let C = (X ×Y ) \O. Then C is a closed set in X ×Y with C ∩(A ×B) = ∅. Let D = C ∩(X ×B). Since D is a closed set in X × B, π0 (D) is closed in X. Since D misses A × B, π0 (D) also misses A. Let P = X \ π0 (D). Then P is an open set in X with A ⊂ P and P × B ⊂ O. By the symmetry, (2) is also true. 2 Recall that a subset S of a space X is normally located in X if for each open set U in X with S ⊂ U , there is an open set V in X with S ⊂ V ⊂ ClX V ⊂ U . Obviously, a space X is normal if and only if each closed set in X is normally located in X. Lemma 3.8. Let X and Y be spaces with A ⊂ X and B ⊂ Y , and let O be an open set in X × Y with A × B ⊂ O. Assume that (i) B is normally located in Y ,

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(ii) there is an open set V in Y with B ⊂ V such that A × V ⊂ O, and (iii) there is an open set W in Y with B ⊂ W such that the projection π : X × ClY W → X is a closed map. Then there is an open rectangle P × Q in X × Y with A ⊂ P , B ⊂ Q and P × Q ⊂ O. Proof. Since B is normally located in Y , there is an open set Q in Y with B ⊂ Q ⊂ ClY Q ⊂ V ∩ W . Then A × ClY Q ⊂ A × V ⊂ O. Since X × ClY Q is a closed set in X × ClY W , the projection π  (X × ClY Q) is a closed map. By applying Lemma 3.7(1), we obtain an open set P in X with A ⊂ P and P × ClY Q ⊂ O. Then P × Q satisfies the required conditions. 2 Lemma 3.9. Let X and Y be spaces such that the projections π0 : X × Y → X and π1 : X × Y → Y are closed maps. Assume that either X or Y is normal. Let A and B be closed sets in X and Y , respectively. Then for each open set O in X × Y with A × B ⊂ O, there is an open rectangle P × Q in X × Y such that A ⊂ P , B ⊂ Q, and P × Q ⊂ O. Proof. Assume that Y is normal. Since A is closed in X, the projection π1  (A × Y ) is a closed map. By Lemma 3.7(2), there is an open set V in Y such that B ⊂ V and A × V ⊂ O. So we see that the conditions (i), (ii) and (iii) in Lemma 3.8 are satisfied. Hence there is an open rectangle P × Q such that A ⊂ P , B ⊂ Q, and P × Q ⊂ O. 2 Theorem 3.10. Let X be a monotonically normal space. If the projection π : X × X → X is a closed map, then X is Δ-paracompact. Proof. By the assumption, it is easy to see that the two projections πi (i = 0, 1) from X × X onto the first coordinate and onto the second coordinate are both closed maps. Let E ∈ S(X, κ) with κ ∈ S ∗ (X). And let   G be an open cover of X. It suffices from Lemma 3.2 to find an open set U of X with eE SE ∩ (γ, κ) ⊂ U for some γ ∈ κ such that: for each x0 , x1 ∈ U , there is a G ∈ G with x0 , x1 ∈ G. Let e = eE and S = SE . By PDL (= the Pressing Down Lemma), there is a γ ∈ κ such that {e[S ∩(γ, α]] : α ∈ (γ, κ)} partially refines G. Put E0 = e[S ∩ (γ, κ)]. Then E0 is a closed set in X such that: for each  x0 , x1 ∈ E0 , there is a G ∈ G with x0 , x1 ∈ G. Let O = {G × G : G ∈ G}. Then O is an open set in X × X and E0 × E0 ⊂ O holds. By applying Lemma 3.9, we obtain an open set U in X with E0 ⊂ U such that  U × U ⊂ O. Pick any x0 , x1 ∈ U . By x0 , x1  ∈ U × U ⊂ O = {G × G : G ∈ G}, there is a G0 ∈ G with x0 , x1  ∈ G0 × G0 . So x0 , x1 ∈ G0 holds. Hence U satisfies the required condition. 2 Recall that if X is a first countable space and Y is a countably compact space, then the projection π : X × Y → X is a closed map. So we immediately have Corollary 3.11. ([2]) Every first-countable, countably compact and monotonically normal space is Δ-paracompact. 4. Δ-normality on monotonically normal spaces Countable tightness Recall the following lemma which is convenient to deal with Δ-normality. Lemma 4.1. ([7, Theorem 2.1]) A space X is Δ-normal if and only if for every open cover G of X, there is an open cover U of X such that 

ClX×X (

{U × U : U ∈ U}) ⊂



{G × G : G ∈ G}.

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Lemma 4.2. For a monotonically normal space X, the following are equivalent: (a) X is Δ-normal.  (b) For each closed set C in X and for each family G of open sets in X with C ⊂ G, there is a family U  of open sets in X with C ⊂ U such that: ClX×X (



{U × U : U ∈ U}) ⊂



{G × G : G ∈ G}.

(c) For each E ∈ S(X, κ) with κ ∈ S ∗ (X) and for each open cover G of X, there is a family U of open sets  in X with eE (SE ∩ (γ, κ)) ⊂ U for some γ ∈ κ such that: ClX×X (



{U × U : U ∈ U}) ⊂



{G × G : G ∈ G}.

 Proof. (a) → (b): Since X is normal, we can take an open set P in X with C ⊂ P ⊂ ClX P ⊂ G. Since G ∪ {X \ ClX P } is an open cover of X, it follows from Lemma 4.1 that there is an open cover V of X such that: ClX×X (

  {V × V : V ∈ V}) ⊂ {G × G : G ∈ G} ∪ [(X \ ClX P ) × (X \ ClX P )].

Let U = {P ∩ V : V ∈ V}. Then it is easily seen that U satisfies the desired condition. (b) → (c): Trivial. (c) → (a): Let G be an open cover of X. Let  V = V : V is an open set in X such that there is a family UV of open sets in X with     V ⊂ UV such that: ClX×X ( {U × U : U ∈ UV }) ⊂ {G × G : G ∈ G} . Then note that V is an open cover of X. By Corollary 2.4, V has a locally finite subcover (open refinement) V0 . Let U = {V ∩ U : V ∈ V0 and U ∈ UV }. Then U is an open cover of X. Pick a point 

x∗ = x0 , x1  ∈ ClX×X (

{U × U : U ∈ U}).

By the choice of V, it suffices to show that there is a V0 ∈ V such that x∗ ∈ ClX×X (



{U × U : U ∈ UV0 }).

Assume the contrary. There is an open neighborhood P of x0 in X such that F = {V ∈ V0 : P ∩ V = ∅} is finite. By the assumption, for each V ∈ F, there is an open neighborhood OV of x∗ in X × X disjoint from   {U × U : U ∈ UV }. Let O = (P × X) ∩ ( V ∈F OV ). Then O is an open neighborhood of x∗ in X × X  disjoint from {U × U : U ∈ U}. This contradicts to the choice of x∗ . 2 Remark 4.3. Monotone normality of X in the assumption of Lemma 4.2 is used only for deriving (c) → (a). It suffices only assuming normality for (a) → (b). We do not need any assumption for (b) → (c). Lemma 4.4. ([9, Lemma 6.2]) Let X and Y be monotonically normal spaces. Let E and F be closed sets in X and Y , respectively, and let O be an open set in X × Y with E × F ⊂ O. If there are two open sets O0 and O1 in X × Y such that E × F ⊂ O0 ∩ O1 , (E × Y ) ∩ ClX×Y O0 ⊂ O and (X × F ) ∩ ClX×Y O1 ⊂ O, then there is an open set W ∗ in X × Y such that E × F ⊂ W ∗ ⊂ ClX×Y W ∗ ⊂ O.

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Definition 4. ([9]) Let κ be a regular cardinal. A space Y has the κ-descending open preserving property at q ∈ Y (the κ-dop property at q ∈ Y for short) if for each descending sequence {Vα : α ∈ κ} of open  neighborhoods of q in Y , there is an open neighborhood W of q in Y with W ⊂ α∈κ Wα . We say that Y has the κ-dop property if it has the κ-dop property at each point of Y . If a space Y has the κ-dop property at each point in F ⊂ Y , then for each descending sequence   {Qξ : ξ ∈ κ} of open sets in Y with F ⊂ ξ∈κ Qξ , we obtain an open set Q in Y with F ⊂ Q ⊂ ξ∈κ Qξ  by letting Q = IntY ( ξ∈κ Qξ ). We define a weakened property of the κ-dop property as below. Definition 5. Let κ be a regular cardinal, and Y a space with F ∈ S(Y, τ ) and τ ∈ S ∗ (Y ). We say that Y has the almost κ-dop property over F if for each descending sequence {Qξ : ξ ∈ κ} of open sets in Y with   eF (SF ∩ (δ0 , τ )) ⊂ ξ∈κ Qξ for some δ0 ∈ τ , there is an open set Q in Y with Q ⊂ ξ∈κ Qξ such that eF (SF ∩ (δ, τ )) ⊂ Q for some δ ∈ τ . Obviously, the following lemma holds. Lemma 4.5. Let κ be a regular cardinal, and Y a space with F ∈ S(Y, τ ) and τ ∈ S ∗ (Y ). If Y has the κ-dop property at each point in F , then Y has the almost κ-dop property over F . Lemma 4.6. Let X and Y be spaces with E ∈ S(X, κ) and F ∈ S(Y, κ), where κ ∈ S ∗ (X) ∩ S ∗ (Y ). Assume that Y has the almost κ-dop property over F . Let O be an open set in X × Y . If ΔE,F = {α ∈ SE ∩ SF : eE (α), eF (α) ∈ O} is stationary in κ, then there are a closed set E  in E and an open set Q in Y with     E  = eE SE ∩ (γ, κ) and eF SF ∩ (γ, κ) ⊂ Q for some γ ∈ κ such that E  × Q ⊂ O. Proof. By PDL, there are a γ0 ∈ κ and a Δ0 ⊂ ΔE,F ∩ (γ0 , κ) which is stationary in κ such that: for each     ξ ∈ Δ0 , there is an open rectangle Uξ × Vξ in X × Y with eE SE ∩ (γ0 , ξ] ⊂ Uξ , eF SF ∩ (γ0 , ξ] ⊂ Vξ and     Uξ × Vξ ⊂ O. Let Eα = Fβ = ∅ for each α, β ≤ γ0 . And let Eα = eE SE ∩ (γ0 , α] and Fβ = eF SF ∩ (γ0 , β] for each α, β ∈ (γ0 , κ). Let Qα = Int{y ∈ Y : Eα × {y} ⊂ O} for each α ∈ κ. Then Eα × Qα ⊂ O. Since {Eα : α ∈ κ} is an ascending sequence, {Qα : α ∈ κ} is a descending sequence of open sets in Y . Let α, β ∈ κ. By taking ξ ∈ Δ0 with α, β ≤ ξ, it follows from Eα ⊂ Eξ ⊂ Uξ that Eα × Vξ ⊂ Uξ × Vξ ⊂ O, so   Fβ ⊂ Fξ ⊂ Vξ ⊂ Qα holds. We have eF (SF ∩ (γ0 , κ)) = β∈κ Fβ ⊂ α∈κ Qα . Since Y has the almost κ-dop  property over F , there is an open set Q in Y with Q ⊂ α∈κ Qα such that eF (SF ∩ (γ, κ)) ⊂ Q for some  γ ∈ κ with γ0 ≤ γ. Let E  = eE (SE ∩ (γ, κ)). Then we have E  × Q ⊂ α∈κ (Eα × Qα ) ⊂ O. 2 Lemma 4.7. Let X be a monotonically normal space. If X has the almost κ-dop property over E for each E ∈ S(X, κ) with κ ∈ S ∗ (X), then X is Δ-normal. Proof. Take an E ∈ S(X, κ) with κ ∈ S ∗ (X) and take an open cover G of X. Let O = Then



{G × G : G ∈ G}.

ΔE,E = {α ∈ SE ∩ SE : eE (α), eE (α) ∈ O} = SE is obviously stationary in κ. Applying Lemma 4.6 twice, we obtain an open rectangle P × Q in X × X with   E  := eE SE ∩ (γ, κ) ⊂ P ∩ Q for some γ ∈ κ such that E  × Q ⊂ O and P × E  ⊂ O. Take an open set N in X with E  ⊂ N ⊂ ClX N ⊂ P ∩ Q. Let O0 = X × N and O1 = N × X. Then O0 and O1 are open in X × X with E  × E  ⊂ O0 ∩ O1 such that (E  × X) ∩ ClX×X O0 ⊂ E  × ClX N ⊂ E  × Q ⊂ O and (X × E  ) ∩ ClX×X O1 ⊂ ClX N × E  ⊂ P × E  ⊂ O.

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Hence it follows from Lemma 4.4 that there is an open set W ∗ in X × X with E  × E  ⊂ W ∗ ⊂ ClX×X W ∗ ⊂ O. Now we put U = {U : U is an open set in X such that U × U ⊂ W ∗ }. Then U covers E  and it satisfies 

ClX×X (

{U × U : U ∈ U}) ⊂ ClX×X W ∗ ⊂



{G × G : G ∈ G}.

Hence it follows from Lemma 4.2 that X is Δ-normal. 2 The following is immediately obtained from Lemmas 4.5 and 4.7. Proposition 4.8. Let X be a monotonically normal space. If X has the κ-dop property for each κ ∈ S ∗ (X), then X is Δ-normal. For a regular cardinal κ, recall that a space Y has the tightness less than κ if for each A ⊂ Y and each y ∈ ClY A, there is some B ⊂ A with |B| < κ and y ∈ ClY B. Lemma 4.9. ([9, Lemma 3.8(2)]) Let κ be a regular cardinal and Y a space with q ∈ Y . If Y has the tightness less than κ, then Y has the κ-dop property at q. In particular, a space X has countable tightness if it has the tightness less than ω1 . Of course, every first countable space has countable tightness. Theorem 4.10. Every monotonically normal space having countable tightness is Δ-normal. Proof. Let X be a monotonically normal space with countable tightness. Take a κ ∈ S ∗ (X). Then X has the tightness less than κ by κ > ω. It follows from Lemma 4.9 that X has the κ-dop property. Hence Proposition 4.8 assures that X is Δ-normal. 2 Countable extent A space X has countable extent if it has no uncountable closed discrete subset in X. Of course, every countably compact space has countable extent. The purpose of this subsection is to give an affirmative answer to [2, Problem 2.2] as follows. Theorem 4.11. Every monotonically normal space having countable extent is Δ-normal. We prove the following general result instead of this, because each κ ∈ S ∗ (X) is an uncountable cardinal. Proposition 4.12. Let X be a monotonically normal space. If X has no closed discrete subset of size κ for each κ ∈ S ∗ (X), then X is Δ-normal. To prove this, it suffices from Lemma 4.7 to show the lemma below. Lemma 4.13. Let Y be a monotonically normal space with F ∈ S(Y, κ), where κ ∈ S ∗ (Y ). If Y does not have a closed discrete subset of size κ, then Y has the almost κ-dop property over F . Proof. Let T = SF and f = eF . And let {Qξ : ξ ∈ κ} be a descending sequence of open sets in Y with   F  = f (T ∩ (γ0 , κ)) ⊂ ξ∈κ Qξ for some γ0 ∈ κ. Put Q = IntY ( ξ∈κ Qξ ). Then Q is an open set in Y    with Q ⊂ ξ∈κ Qξ . To show that f T ∩ (γ, κ) ⊂ Q for some γ ∈ κ, it suffices from PDL to show that f −1 (Q) = {ζ ∈ T : f (ζ) ∈ Q} is stationary in κ.

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Assume that f −1 (Q) is non-stationary in κ. Since Y is monotonically normal, we can take an ascending     sequence {Mξ : ξ ∈ κ} of open sets in Y such that f T ∩ [0, ξ] ⊂ Mξ ⊂ ClY Mξ ⊂ Y \ f T ∩ (ξ, κ) for each   ξ ∈ κ. In fact, we may let Mξ = M f (T ∩ [0, ξ]), f (T ∩ (ξ, κ)) , where M is a monotone normality operator  for Y (see Definition 3). By normality of Y , take an open set N in Y with F ⊂ N ⊂ ClY N ⊂ ξ∈κ Mξ . Moreover, take a monotonically normal assignment H for Y (see Lemma 2.1). By induction on α ∈ κ, we take strictly increasing sequences {ξ(α) : α ∈ κ} ⊂ [γ0 , κ), {ζ(α) : α ∈ κ} ⊂ T \ f −1 (Q) and a sequence {yα : α ∈ κ} ⊂ N ⊂ Y , satisfying that for each α ∈ κ, (i) α ≤ ξ(α) < ζ(α) < ξ(α + 1),   (ii) yα ∈ H f (ζ(α)), Mζ(α) \ ClY Mξ(α) ∩ N , / Qξ(α+1) . (iii) yα ∈ At first, put ξ(0) = γ0 . Let α ∈ κ and assume that ξ(α) ∈ [γ0 , κ) is determined. Take a ζ(α) ∈ T \ f −1 (Q)  / Q = IntY ( ξ∈κ Qξ ) that with ξ(α) < ζ(α). It follows from f (ζ(α)) ∈   N ∩ H f (ζ(α)), Mζ(α) \ ClY Mξ(α) ⊂ Qξ , ξ∈κ

  so we can take a yα ∈ N ∩ H f (ζ(α)), Mζ(α) \ ClY Mξ(α) and ξ(α + 1) ∈ κ with ζ(α) < ξ(α + 1) and yα ∈ / Qξ(α+1) . Next let α ∈ κ be a limit ordinal, and assume that ξ(β) is determined for each β < α. Then let ξ(α) = sup{ξ(β) : β < α}. Thus (i)–(iii) are satisfied. After finishing induction, put D = {yα : α ∈ κ}. If α0 < α1 < κ, then we have yα0 ∈ Mζ(α0 ) ⊂ Mξ(α1 ) ⊂ ClY Mξ(α1 ) by ζ(α0 ) < ξ(α0 + 1) ≤ ξ(α1 ), and we have yα1 ∈ / ClY Mξ(α1 ) by (ii). So we obtain yα0 = yα1 . Therefore |D| = κ holds. Pick any y ∈ Y . We will find an open neighborhood V of y in Y with |V ∩ D| ≤ 1. In case y ∈ / ClY N , V = Y \ ClY N is a required one, because V ∩ D ⊂ V ∩ N = ∅. So we assume that y ∈ ClY N . Then  y ∈ Mβ ⊂ Mξ(β) ⊂ ClY Mξ(β) holds for some β ∈ κ since ClY N ⊂ ξ∈κ Mξ and α ≤ ξ(α) holds for each α ∈ κ. Take the least β ∈ κ such that y ∈ ClY Mξ(β) . Then y ∈ / Mζ(α) \ ClY Mξ(α) for any α ∈ κ except the case of β = α + 1. Actually, y ∈ ClY Mξ(β) ⊂ ClY Mξ(α) for each α with β ≤ α < κ by ξ(β) ≤ ξ(α). And y∈ / Mζ(α) for any α < β with α + 1 = β by the minimality of β, because by ζ(α) < ξ(α + 1) < ξ(β), we have y ∈ / ClY Mξ(α+1) ⊃ Mξ(α+1) ⊃ Mζ(α) . In case y ∈ / F  , put V = H(y, Y \ F  ). Then V is an open neighborhood of y in Y . For each α ∈ κ,  f (ζ(α)) ∈ F holds by ζ(α) ∈ T and γ0 ≤ ξ(α) < ζ(α). For any α ∈ κ with β = α + 1, it follows from y∈ / Mζ(α) \ ClY Mξ(α) and f (ζ(α)) ∈ / Y \ F  that   H(y, Y \ F  ) ∩ H f (ζ(α)), Mζ(α) \ ClY Mξ(α) = ∅,   so yα ∈ H f (ζ(α)), Mζ(α) \ ClY Mξ(α) does not belong to V = H(y, Y \ F  ). Hence we obtain |V ∩ D| ≤ 1.  In case y ∈ F  , put V = Qξ(β) ∩ Mξ(β) . We have y ∈ F  ⊂ ξ∈κ Qξ ⊂ Qξ(β) and y ∈ F ∩ ClY Mξ(β) ⊂     f (T ) \ f T ∩ (ξ(β), κ) = f T ∩ [0, ξ(β)] ⊂ Mξ(β) . So V is an open neighborhood of y in Y . We see that   yα ∈ / V for each α ∈ κ with β ≤ α since yα ∈ M f (ζ(α)), Mζ(α) \ ClY Mξ(α) ) ⊂ Mζ(α) \ ClY Mξ(α) does not belong to V ⊂ Mξ(β) ⊂ Mξ(α) ⊂ ClY Mξ(α) by ξ(β) ≤ β(α). We also see that yα ∈ / V for any α < β since yα ∈ / Qξ(α+1) does not belong to V ⊂ Qξ(β) ⊂ Qξ(α+1) by α + 1 ≤ β and ξ(α + 1) ≤ ξ(β). Hence V misses D. Thus, it has been seen that for each y ∈ Y , there is an open neighborhood V of y in Y with |V ∩ D| ≤ 1. Hence D is a closed discrete subset in Y of size κ. This contradicts the assumption of Y . 2 Thus, Proposition 4.12 has been proved. As a consequence, Theorem 4.11 is true.

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5. A neighborhood property for Δ-paracompactness The κ-nbdΔ property Definition 6. Let κ be a regular cardinal. Let Y be a space with q ∈ Y . We say that Y has the κ-nbdΔ property at q if for each sequence {Vα : α ∈ κ} of open neighborhoods of q in Y , there is an open neighborhood W of q in Y such that: for each y0 , y1 ∈ W , {α ∈ κ : y0 , y1 ∈ Vα } is unbounded in κ. We also say that Y has the κ-nbdΔ property if it has the κ-nbdΔ property at each point of Y . Next, we recall the two neighborhood properties dealt with in [9] and consider their implications with this new one. Let κ be a regular cardinal. A space Y has orthocaliber κ at q ∈ Y [9,11] if for each sequence {Vα : α ∈ κ} of open neighborhoods of q in Y , there is an open neighborhood W of q in Y such that {α ∈ κ : W ⊂ Vα } is unbounded in κ. We say that Y has orthocaliber κ if it has orthocaliber κ at each point of Y . Lemma 5.1. Let κ be a regular cardinal. Let Y be a space with q ∈ Y . (1) If Y has orthocaliber κ at q, then it has the κ-nbdΔ property at q. (2) If Y has the κ-nbdΔ property at q, then it has the κ-dop property at q. Proof. Let V = {Vα : α ∈ κ} be a sequence of open neighborhoods of q in Y . (1): There is an open neighborhood W of q in Y such that W ⊂ Vα for κ many α’s. It is obvious that W witnesses the κ-nbdΔ property at q for Y . (2): Assume that V is descending. Let W be an open neighborhood of q in Y which witnesses the κ-nbdΔ property at q for V. Pick any y ∈ W and take any α ∈ κ. Considering as y0 := y, y1 := q ∈ W , there is δ ∈ κ  with α ≤ δ and y0 , y1 ∈ Vδ . So we have y = y0 ∈ Vδ ⊂ Vα , because V is descending. Hence W ⊂ α∈κ Vα holds. 2 Lemma 5.2. Let κ be a regular cardinal and Y a space with q ∈ Y . The following are equivalent: (a) Y has the κ-nbdΔ property at q. (b) For each sequence {Vα : α ∈ κ} of open neighborhoods of q in Y , there is a descending sequence {Wα : α ∈ κ} of open neighborhoods of q in Y such that: for each y0 ∈ Wα0 and y1 ∈ Wα1 with α0 , α1 ∈ κ, there is a δ ∈ κ with α0 , α1 ≤ δ and y0 , y1 ∈ Vδ . (c) For each sequence {Vα : α ∈ κ} of open neighborhoods of q in Y , there is a sequence {Wα : α ∈ κ} of open neighborhoods of q in Y such that: for each y0 ∈ Wα0 and y1 ∈ Wα1 with α0 , α1 ∈ κ, there is a δ ∈ κ with α0 , α1 ≤ δ and y0 , y1 ∈ Vδ . (d) Y × Y has the κ-dop property at q, q. Proof. Let V = {Vα : α ∈ κ} be a sequence of open neighborhoods of q in Y , except the last (a) → (d). (a) → (b): Let W be an open neighborhood of q in Y witnessing the definition of (a) for V. Let Wα = W for each α ∈ κ. Then it is obvious that {Wα : α ∈ κ} satisfies the condition (b). (b) → (c): This is trivial. (c) → (b): Let {Wβ : β ∈ κ} be a sequence of neighborhoods of q in Y witnessing the definition of (c)  for V. Put Wα = β∈κ\α Wβ for each α ∈ κ. Then W := {Wα : α ∈ κ} is a descending sequence of open neighborhoods of q in Y . Let y0 ∈ Wα0 and y1 ∈ Wα1 with α0 , α1 ∈ κ. Then there are a β0 ∈ κ \ α0 and a β1 ∈ κ \ α1 such that y0 ∈ Wβ 0 and y1 ∈ Wβ 1 . And there is δ ∈ κ with β0 , β1 ≤ δ and y0 , y1 ∈ Vδ . We have αi ≤ βi ≤ δ by βi ∈ κ \ αi for i = 0, 1. Hence W satisfies the condition (b).

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(b) → (a): Let W = {Wβ : β ∈ κ} be a descending sequence of open neighborhoods of q in Y witnessing (b) for V. First, we show that Y has the κ-dop property at q. For that, assume that V is descending. Pick any y ∈ W0 and take an α ∈ κ. Considering as y0 := y ∈ W0 and y1 := q ∈ Wα , there is γ ∈ κ with  α ≤ γ and y0 , y1 ∈ Vγ . Then we have y = y0 ∈ Vγ ⊂ Vα , because V is descending. Hence W0 ⊂ α∈κ Vα holds, which means that Y has the κ-dop property at q. Next we show that (a) holds. Since W is descending,  by the κ-dop property, there is an open neighborhood W of q with W ⊂ α∈κ Wα . Pick any y0 , y1 ∈ W and any β ∈ κ. Considering as α0 = α1 = β, since yi ∈ W ⊂ Wαi = Wβ (i = 0, 1), there is δ ∈ κ with β = αi ≤ δ (i = 0, 1) and y0 , y1 ∈ Vδ . Hence {α ∈ κ : y0 , y1 ∈ Vα } is unbounded in κ.  (d) → (a): For each ξ ∈ κ, put Vξ∗ = α∈κ\ξ (Vα × Vα ). Then {Vξ∗ : ξ ∈ κ} is a descending sequence of open neighborhoods of q, q in Y × Y . There is an open neighborhood W ∗ of q, q in Y × Y such that  W ∗ ⊂ ξ∈κ Vξ∗ . Take an open neighborhood W of q in Y such that W ×W ⊂ W ∗ . Pick any y0 , y1 ∈ W . Take any ξ ∈ κ. Then we have y0 , y1  ∈ W × W ⊂ W ∗ ⊂ Vξ∗ , so there is δ ∈ κ \ ξ such that y0 , y1  ∈ Vδ × Vδ . Then we have δ ∈ κ with ξ ≤ δ and y0 , y1 ∈ Vδ . Hence {α ∈ κ : y0 , y1 ∈ Vα } is unbounded in κ. (a) → (d): Let {Vξ∗ : ξ ∈ κ} be a descending sequence of open neighborhoods of q, q in Y × Y . Take an open neighborhood Vα of q in Y such that Vα ×Vα ⊂ Vα∗ for each α ∈ κ. Then there is an open neighborhood W of q in Y such that: for each y0 , y1 ∈ W , there are unbounded many α ∈ κ such that y0 , y1 ∈ Vα . Then W ∗ := W × W is an open neighborhood of q, q in Y × Y . Pick any y0 , y1  ∈ W ∗ . Take any ξ ∈ κ. By y0 , y1 ∈ W , there is δ ∈ κ with ξ ≤ δ and y0 , y1 ∈ Vδ . Then we have y0 , y1  ∈ Vδ × Vδ ⊂ Vδ∗ ⊂ Vξ∗ . So   y0 , y1  ∈ ξ∈κ Vξ∗ holds. Hence we obtain W ∗ ⊂ ξ∈κ Vξ∗ . 2 Normality of Δ-paracompact products In this subsection, we show that Δ-paracompactness implies normality in the class of all products of a monotonically normal space with a compact space. Lemma 5.3. Let S be a stationary subset of a regular uncountable cardinal κ. Let Y be a regular space. If S × Y is Δ-paracompact, then Y has the κ-nbdΔ property. Proof. Let V = {Vα : α ∈ κ} be a sequence of open neighborhoods of q ∈ Y . It suffices to show that there is a sequence {Wα : α ∈ κ} of open neighborhoods of q ∈ Y , satisfying Lemma 5.2(c) for V. Let Sα = S ∩ [0, α] for each α ∈ κ. Claim. There is an open neighborhood Q of q in Y such that S × Q ⊂



α∈κ (Sα

× Vα ).

Proof. Applying Δ-paracompactness for an open cover {Sα × Vα : α ∈ κ} ∪ {S × (Y \ {q})} of S × Y , we obtain a locally finite open cover U0 of S × Y such that: for each z0 , z1 ∈ U0 with U0 ∈ U0 , if z1 ∈ S × {q}, then there is an α ∈ κ with z0 , z1 ∈ Sα × Vα . Since U0 is point countable, we can take a U0 ∈ U0 and a γ0 ∈ κ such that (S ∩ (γ0 , κ)) × {q} ⊂ U0 . Take an α0 ∈ S ∩ (γ0 , κ). By α0 , q ∈ U0 , there  is an open neighborhood Q of q in Y such that {α0 } × Q ⊂ U0 . We show that S × Q ⊂ α∈κ (Sα × Vα ). Pick any β, y ∈ S × Q. Let z0 = α0 , y. Then we have z0 ∈ {α0 } × Q ⊂ U0 . Take an α1 ∈ S ∩ (γ0 , κ) with β ≤ α1 . Let z1 = α1 , q. Then we have z1 ∈ (S ∩ (γ0 , κ)) × {q} ⊂ U0 . By z0 , z1 ∈ U0 with U0 ∈ U and z1 ∈ S × {q}, there is δ ∈ κ with z0 , z1 ∈ Sδ × Vδ . Since α1 ∈ Sδ by z1 ∈ Sδ × Vδ , we have β ≤ α1 ≤ δ,  thus β ∈ Sδ . Since y ∈ Vδ by z0 ∈ Sδ × Vδ , we have β, y ∈ Sδ × Vδ . Therefore S × Q ⊂ α∈κ (Sα × Vα ) holds. 2

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Take an open neighborhood N of q in Y with ClY N ⊂ Q. Applying Δ-paracompactness for an open cover {Sα × Vα : α ∈ κ} ∪ {S × (Y \ ClY N )} of S × Y , we obtain a locally finite open cover U1 of S × Y such that: for each z0 , z1 ∈ U1 ∩ (S × N ) with U1 ∈ U1 , there is an α ∈ κ with z0 , z1 ∈ Sα × Vα . Since U1 is point countable, we can take a U1 ∈ U1 and a γ1 ∈ κ such that (S ∩ (γ1 , κ)) × {q} ⊂ U1 . For each α ∈ κ, take a ξ(α) ∈ S ∩ (γ1 , κ) with α ≤ ξ(α). Then ξ(α), q ∈ U1 , so we can take an open neighborhood Wα of q in Y such that Wα ⊂ N and {ξ(α)} ×Wα ⊂ U1 . Now, we show that {Wα : α ∈ κ} satisfies Lemma 5.2(c) for V. Let y0 ∈ Wα0 and y1 ∈ Wα1 with α0 , α1 ∈ κ. Let z0 = ξ(α0 ), y0  and z1 = ξ(α1 ), y1 . Then z0 , z1 ∈ U1 ∩ (S × N ) holds with U1 ∈ U1 , so there is δ ∈ κ with z0 , z1 ∈ Sδ × Vδ . By ξ(α0 ), ξ(α1 ) ∈ Sδ , we have α0 ≤ ξ(α0 ) ≤ δ and α1 ≤ ξ(α1 ) ≤ δ. Therefore we obtain δ ∈ κ with α0 , α1 ≤ δ such that y0 , y1 ∈ Vδ . Hence Y has the κ-nbdΔ property at q. 2 Since a space which is homeomorphic to a closed subspace of a Δ-paracompact space is also Δ-paracompact, Lemma 5.3 immediately yields Corollary 5.4. Let X and Y be regular spaces. If X × Y is Δ-paracompact, then • Y has the κ-nbdΔ property for each κ ∈ S ∗ (X). • X has the κ-nbdΔ property for each κ ∈ S ∗ (Y ). Now, we should recall the following result. Theorem 5.5. ([16], [9, Theorem 4.1]) Let X be a monotonically normal space and K a compact space. Then the following are equivalent: (a) (b) (c) (d)

X × K is normal. E × K is normal for each E ∈ S(X). K has the κ-dop property for each κ ∈ S ∗ (X). K has the SE -docs property for each E ∈ S(X).

The SE -docs property, appearing in (d), is a neighborhood property defined in [9], and the definition is also described in the next section. Corollary 5.6. Let X be a monotonically normal space and K a compact space. If X × K is Δ-paracompact, then X × K is normal. Proof. Assume that X ×K is Δ-paracompact. It follows from Corollary 5.4 that K has the κ-nbdΔ property for each κ ∈ S ∗ (X). By Lemma 5.1(2), K has the κ-dop property for each κ ∈ S ∗ (X). Therefore it follows from Theorem 5.5 that X × K is normal. 2 Δ-paracompactness of non-normal spaces For a regular uncountable cardinal κ, let Zκ = {α, β ∈ (κ + 1) × (κ + 1) : α ≤ β} \ {κ, κ} be a subspace of (κ +1) ×(κ +1). It is known that Zκ is not normal. Hart stated in [7, Example 3.5] that Zω1 is not Δ-normal. On the other hand, Buzyakova proved in [3, Example 2.12] that Zω1 is Δ-paracompact. However, the following lemma contradicts the latter result.

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Lemma 5.7. For each regular uncountable cardinal κ, the space Zκ is neither Δ-paracompact nor Δ-normal. Proof. Let κ be a regular uncountable cardinal. Let G0 (ξ) = Zκ ∩ ([0, ξ] × [0, ξ]) and G1 (ξ) = Zκ ∩ ([0, ξ] × (ξ, κ]) for each ξ ∈ κ. Let G = {G0 (ξ) : ξ ∈ κ} ∪ {G1 (ξ) : ξ ∈ κ}. Then G is an open cover of Zκ . Assume that Zκ is Δ-paracompact. By Lemma 3.1, there is a locally finite open cover U of Zκ such that for each z0 , z1 ∈ U with U ∈ U, there is a G ∈ G with z0 , z1 ∈ G. Since {{α ∈ κ : α, κ ∈ U } : U ∈ U} is a locally finite open cover of κ, there are γ ∈ κ and U0 ∈ U with (γ, κ) × {κ} ⊂ U0 . Take an α0 ∈ (γ, κ). Since U0 is an open set in Zκ with α0 , κ ∈ U0 , there is β0 ∈ (γ, κ) with z0 := α0 , β0  ∈ U0 . Take an α1 ∈ (β0 , κ). Then z1 := α1 , κ ∈ U0 by γ < β0 < α1 < κ. Since z0 , z1 ∈ U0 with U0 ∈ U, there is a G ∈ G with z0 , z1 ∈ G. And G = G0 (ξ) or G = G1 (ξ) for some ξ ∈ κ. The former cannot happen since z1 = α1 , κ ∈ / Zκ ∩ ([0, ξ] × [0, ξ]) = G0 (ξ) by ξ < κ. The latter also cannot happen. Otherwise ξ < β0 holds by α0 , β0  = z0 ∈ G = G1 (ξ) = Zκ ∩ ([0, ξ] × (ξ, κ]), α1 ≤ ξ holds by α1 , κ = z1 ∈ G = G1 (ξ) = Zκ ∩ ([0, ξ] × (ξ, κ]), and β0 < α1 holds by α1 ∈ (β0 , κ). We have ξ < β0 < α1 ≤ ξ, which is a contradiction. Hence Zκ is not Δ-paracompact. Assume that Zκ is Δ-normal. By Lemma 4.1, there is an open cover U of Zκ such that 

ClZκ ×Zκ (

{U × U : U ∈ U}) ⊂



{G × G : G ∈ G}.

 For each α ∈ κ, there is Uα ∈ U with α, κ ∈ Uα . Take a γα < α and a δα < κ with Zκ ∩ (γα , α] ×  (δα , κ] ⊂ Uα . By PDL, there are γ ∈ κ and S ⊂ κ such that S is stationary in κ and γα = γ for each α ∈ S. Let C = {β ∈ κ : α < β implies δα < β}. Then C is a club set in κ. Choose an α0 ∈ κ with γ < α0 . And choose a β0 ∈ S ∩ Lim(S) ∩ C with α0 < β0 . Let z0 = α0 , β0  and z1 = β0 , κ. Then z0 , z1 ∈ Zκ . For each α ∈ S ∩ (α0 , β0 ), it follows from γ = γα < α0 < α and δα < β0 < κ that   {z0 , α, κ} ⊂ Zκ ∩ (γα , α] × (δα , κ] ⊂ Uα ∈ U. So we have z0 ∈

 {Uα : α ∈ S ∩ (α0 , β0 )} and z1 ∈ ClZκ {α, κ : α ∈ S ∩ (α0 , β0 )} ⊂ ClZκ



 {Uα : α ∈ S ∩ (α0 , β0 )} .

Hence we obtain

 z0 , z1  ∈ ClZκ ×Zκ { z0 , α, κ : α ∈ S ∩ (α0 , β0 )}   ⊂ ClZκ ×Zκ {Uα × Uα : α ∈ S ∩ (α0 , β0 )}    {U × U : U ∈ U} ⊂ {G × G : G ∈ G}. ⊂ ClZκ ×Zκ There is G∗ ∈ G with z0 , z1 ∈ G∗ . Since z1 ∈ / G0 (ξ) for each ξ ∈ κ, there is ζ ∈ κ with G∗ = G1 (ζ).   ∗ Since z0 = α0 , β0  ∈ G = G1 (ζ) = Zκ ∩ [0, ζ] × (ζ, κ] , we obtain α0 ≤ ζ < β0 . However, since   z1 = β0 , κ ∈ G∗ = G1 (ζ) = Zκ ∩ [0, ζ] × (ζ, κ] , we obtain β0 ≤ ζ. This is a contradiction. 2 As a Δ-paracompact space which is not normal, we give the following example.   Example 5.8. For two regular uncountable cardinals κ and τ , let Xκ,τ = (κ + 1) × (τ + 1) \ {κ, τ } be a subspace of (κ + 1) × (τ + 1). Then the space Xκ,τ is Δ-paracompact for any distinct regular uncountable cardinals κ and τ .

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Proof. Let κ and τ be two regular uncountable cardinals with κ = τ . For each γ ∈ κ and δ ∈ τ , put U0 (γ, δ) = (γ, κ) × (δ, τ ], U1 (γ, δ) = (γ, κ] × (δ, τ ), K0 (γ) = [0, γ] × [0, τ ], K1 (δ) = [0, κ] × [0, δ]. Then {U0 (γ, δ), U1 (γ, δ), K0 (γ), K1 (δ)} is an open cover of Xκ,τ . Define two collections U0 (γ, δ) and U1 (γ, δ) of open sets of Xκ,τ by U0 (γ, δ) = {(γ, α] × (δ, τ ] : α ∈ (γ, κ)}, U1 (γ, δ) = {(γ, κ] × (δ, β] : β ∈ (δ, τ )}.   Then we have U0 (γ, δ) = U0 (γ, δ), U1 (γ, δ) = U1 (γ, δ). Let F0 = κ × {τ } and F1 = {κ} × τ . Then F0 and F1 are disjoint closed subsets of Xκ,τ . Claim 1. For each family G0 of open sets in Xκ,τ with F0 ⊂ U0 (γ0 , δ0 ) partially refines G0 .



G0 , there are γ0 ∈ κ and δ0 ∈ τ such that

 Proof. For each α ∈ κ \ {0}, α, τ  ∈ F0 ⊂ G0 holds. Take a G0 (α) ∈ G0 with α, τ  ∈ G0 (α). And take γ0 (α) < α and δ0 (α) < τ with (γ0 (α), α] × (δ0 (α), τ ] ⊂ G0 (α). By PDL, we obtain γ0 ∈ κ and S0 ⊂ κ \ {0} which is stationary in κ such that γ0 (α) = γ0 for each α ∈ S0 . Moreover, by κ = τ , there are δ0 ∈ τ and S ⊂ S0 which is stationary in κ such that δ0 (α) ≤ δ0 for every α ∈ S. Then γ0 and δ0 satisfy the required condition. Actually, let α ∈ (γ0 , κ). By taking α0 ∈ S with α ≤ α0 , we have (γ0 , α] × (δ0 , τ ] ⊂ (γ0 (α0 ), α0 ] × (δ0 (α0 ), τ ] ⊂ G0 (α0 ) ∈ G0 . 2 Similarly, we obtain the claim below. Claim 2. For each family G1 of open sets of Xκ,τ with F1 ⊂ U1 (γ1 , δ1 ) partially refines G1 .



G1 , there are γ1 ∈ κ and δ1 ∈ τ such that

It remains to show that Xκ,τ is Δ-paracompact. Let G be an open cover of Xκ,τ . By applying Claim 1 for G0 = G, we obtain γ0 ∈ κ and δ0 ∈ τ such that U0 (γ0 , δ0 ) partially refines G0 = G. By applying Claim 2 for G1 = G, we obtain γ1 ∈ κ and δ1 ∈ τ such that U1 (γ1 , δ1 ) partially refines G1 = G. Take γ ∈ κ and δ ∈ τ with max{γ0 , γ1 } ≤ γ and max{δ0 , δ1 } ≤ δ. Since K0 (γ) and K1 (δ) are compact, we can take a finite  subfamily K of G such that K0 (γ) ∪ K1 (δ) ⊂ K. And let U = {U0 (γ, δ), U1 (γ, δ)} ∪ K. Then U is a finite open cover of Xκ,τ . Let x0 , x1 ∈ U with U ∈ U. It suffices to find a G ∈ G with x0 , x1 ∈ G. If U ∈ K, then U ∈ G, so G = U is a required one. If U = U0 (γ, δ) = (γ, κ) × (δ, τ ], then we can take an α0 ∈ (γ, κ) with x0 , x1 ∈ (γ, α0 ] × (δ, τ ]. By γ0 ≤ γ < α0 and δ0 ≤ δ, we have α0 ∈ (γ0 , κ) and x0 , x1 ∈ (γ0 , α0 ] × (δ0 , τ ] ⊂ G for some G ∈ G since U0 (γ0 , δ0 ) partially refines G. If U = U1 (γ, δ) = (γ, κ] × (δ, τ ), then we can take a β1 ∈ (δ, τ ) with x0 , x1 ∈ (γ, κ] × (δ, β1 ]. By γ1 ≤ γ and δ1 ≤ δ < β1 , we have β1 ∈ (δ1 , τ ) and x0 , x1 ∈ (γ1 , κ] × (δ1 , β1 ] ⊂ G for some G ∈ G since U1 (γ1 , δ1 ) partially refines G. 2 It is well known that each Xκ,τ is not normal. By Lemma 5.7, Xκ,κ is not Δ-normal since it contains Zκ as a closed subspace. So it follows from [3, Corollary 2.7] and Example 5.8 that Xκ,τ is not Δ-normal without regard to κ = τ or κ = τ . Hence we immediately obtain Corollary 5.9. Δ-paracompactness implies neither normality nor Δ-normality. Remark 5.10. Buzyakova stated this result in [3, Lemma 2.13], as a consequence of Zω1 being Δ-paracompact.

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6. Another neighborhood property for Δ-normality The weak S-nbdΔ-property As a weakening of the κ-nbdΔ property, we will define a property called the weak S-nbdΔ property below. Recall Lemma 5.2(a) ↔ (d) which says that Y has the κ-nbdΔ property at q ∈ Y if and only if Y × Y has the κ-dop property at q, q. A sequence {Fα : α ∈ S} of subsets in a space Y , where S is a set of ordinals, is continuously descending  [9] if Fα ⊃ Fα for each α, α ∈ S with α < α and Fα = {Fβ : β ∈ S ∩ α} for each α ∈ S ∩ Lim(S). Definition 7. ([9]) Let Y be a space and S a subset in an ordinal. We say that Y has the S-descending open continuously shrinking property (the S-docs property for short) at q ∈ Y if for each descending sequence V = {Vα : α ∈ S} of open neighborhoods of q in Y , there is a continuously descending sequence F = {Fα : α ∈ S} of closed neighborhoods of q in Y such that Fα ⊂ Vα for each α ∈ S. We call such F a continuous shrinking of V by closed neighborhoods of q. We say that Y has the S-docs property if it has the S-docs property at each point of Y . Definition 8. Let S be a set of ordinals, and Y a space with q ∈ Y . We say that Y has the weak S-nbdΔ property at q if Y × Y has the S-docs property at q, q. We say that Y has the weak S-nbdΔ property if it has the weak S-nbdΔ property at each point of Y . Lemma 6.1. ([9, Lemma 3.4(2)]) Let κ be a regular cardinal and Y a regular space with q ∈ Y . If Y has the κ-dop property at q, then Y has the S-docs property at q for each unbounded subset S of κ. Lemma 6.2. Let Y be a regular space with q ∈ Y . (1) Let S be an unbounded subset in a regular cardinal κ. If Y has the κ-nbdΔ property at q, then Y has the weak S-nbdΔ property at q. (2) Let S be a set of ordinals. If Y has the weak S-nbdΔ property at q, then Y has the S-docs property at q. Proof. (1) It follows from Lemma 5.2 that Y × Y has the κ-dop property at q, q. By Lemma 6.1, Y × Y has the S-docs property at q, q. That is, Y has the weak S-nbdΔ property at q. (2) Assume that Y × Y has the S-docs property at q, q. Let {Vξ : ξ ∈ S} be a descending sequence of open neighborhoods of q in Y . Then {Vξ × Y : ξ ∈ S} is a descending sequence of open neighborhoods of q, q in Y × Y . So there is a continuously descending sequence {Fξ∗ : ξ ∈ S} of closed neighborhoods of q, q in Y × Y such that Fξ∗ ⊂ Vξ × Y for each ξ ∈ S. Let Fξ = {y ∈ Y : y, q ∈ Fξ∗ } for each ξ ∈ S. Then {Fξ : ξ ∈ S} is a continuously descending sequence of closed neighborhoods of q in Y such that Fξ ⊂ Vξ for each ξ ∈ S. Hence Y has the S-docs property at q. 2 We have the following implications for neighborhood properties: orthocaliber κ

⇒

tightness < κ

⇒

κ-nbdΔ ⇓ κ-dop

⇒ ⇒

weak S-nbdΔ ⇓ S-docs

Lemma 6.3. ([9, Fact 3.1 and Lemma 3.2(2)]) Let Y be a space with q ∈ Y . (1) If S is an unbounded subset of a limit ordinal λ and Y has the T -docs property at q for some T with S ⊂ T ⊂ λ, then Y has the S-docs property. (2) If S is a stationary subset of a regular uncountable cardinal κ, and Y has the (S ∩ C)-docs property at q for some club set C of κ, then Y has the S-docs property at q.

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So the following is immediately obtained. Lemma 6.4. Let Y be a space with q ∈ Y . (1) If S is an unbounded subset of a limit ordinal λ and Y has the weak T -nbdΔ property at q for some T with S ⊂ T ⊂ λ, then Y has the weak S-nbdΔ property at q. (2) If S is a stationary subset of a regular uncountable cardinal κ, and Y has the weak (S ∩C)-nbdΔ property at q for some club set C of κ, then Y has the weak S-nbdΔ property at q. Although the following lemma looks to be almost same as Lemma 6.3(2), it is precisely a stronger statement since we may choose a club set C after {Vα : α ∈ S} is given. And it is sometimes useful for deriving the S-docs property. Lemma 6.5. ([9, Lemma 3.2(1)]) Let S be a stationary subset of a regular uncountable cardinal κ and Y a space with q ∈ Y . Then Y has the S-docs property at q if and only if (†) for each descending sequence {Vξ : ξ ∈ S} of neighborhoods of q in Y , there is a club set C of κ such that {Vξ : ξ ∈ S ∩ C} has a continuous shrinking by closed neighborhoods of q in Y . Lemma 6.6. Let Y be a space with q ∈ Y . (1) Let S be a subset of an ordinal λ. Then Y has the weak S-nbdΔ property at q if and only if the property (∗) below holds: (∗) for each sequence {Vα : α ∈ S} of open neighborhoods of q in Y , there is a sequence {Wβ : β ∈ S} of neighborhoods of q in Y such that: for each ξ ∈ S,

ClY ×Y (

γ∈ξ∪{−1}



(Wβ × Wβ )) ⊂

β∈S∩(γ,λ)



(Vα × Vα ).

α∈S\ξ

(2) Let S be a stationary subset in a regular uncountable cardinal κ. Then Y has the weak S-nbdΔ property at q if and only if the property (∗∗) below holds: (∗∗) for each sequence {Vα : α ∈ S} of neighborhoods of q in Y , there are a club set C in κ and a sequence {Wβ : β ∈ S ∩ C} of open neighborhoods of q in Y such that: for each ξ ∈ S ∩ C,

ClY ×Y



γ∈ξ∪{−1}



  (Wβ × Wβ ) ⊂ (Vα × Vα ).

β∈S∩C∩(γ,κ)

α∈S\ξ

Proof. It is obvious that (∗) implies (∗∗) in case S is a stationary subset in a regular uncountable cardinal κ by considering λ = C = κ. Assuming that Y × Y has the S-docs property at q, q, we prove the property (∗). Let {Vα : α ∈ S} be a  sequence of open neighborhoods of q in Y . Let Vξ∗ = α∈S\ξ (Vα × Vα ) for each ξ ∈ κ. Then {Vξ∗ : ξ ∈ S} is a descending sequence of open neighborhoods of q, q in Y × Y , so it has a continuous shrinking {Fξ∗ : ξ ∈ S} by closed neighborhoods of q, q in Y × Y . For each β ∈ S, take an open neighborhood Wβ of q in Y such that Wβ × Wβ ⊂ Fβ∗ . To see that {Wβ : β ∈ S} is a required sequence for the property (∗), let ξ ∈ S and y∗ ∈

γ∈ξ∪{−1}

ClY ×Y





 (Wβ × Wβ ) .

β∈S∩(γ,λ)

It suffices to show that y ∗ ∈ Fξ∗ holds because Fξ∗ ⊂ Vξ∗ =



α∈S\ξ (Vα

× Vα ).

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In case ξ ∈ S \ Lim(S): Take γ < ξ with S ∩ (γ, ξ) = ∅. Then we have y ∗ ∈ ClY ×Y (





(Wβ × Wβ )) ⊂ ClY ×Y (

β∈S∩[ξ,λ)

Fβ∗ ) = Fξ∗ .

β∈S∩[ξ,λ)

In case ξ ∈ S ∩ Lim(S): Take any ζ ∈ S ∩ ξ. Then we have y ∗ ∈ ClY ×Y





  (Wβ × Wβ ) ⊂ ClY ×Y

β∈S∩(ζ,λ)



 Fβ∗ ⊂ ClY ×Y Fζ∗ = Fζ∗ .

β∈S∩(ζ,λ)

 Hence we obtain y ∗ ∈ ζ∈S∩ξ Fζ∗ = Fξ∗ . Next assume that the property (∗) holds. Or the property (∗∗) holds where S is a stationary subset of a regular uncountable cardinal κ = λ. We show that Y × Y has the S-docs property at q, q. Let {Vξ∗ : ξ ∈ S} be a descending sequence of neighborhoods of q, q in Y × Y . Take an open neighborhood Vα of q in Y such that Vα × Vα ⊂ Vα∗ for each α ∈ S. Applying the property (∗) or (∗∗), we obtain a sequence {Wβ : β ∈ S0 } of open neighborhoods of q in Y , where S0 = S ⊂ λ or S0 = S ∩ C ⊂ κ = λ for a club set C in κ when S is a stationary subset of a regular uncountable cardinal κ, such that: for each ξ ∈ S0 ,

ClY ×Y





   (Wβ × Wβ ) ⊂ (Vα × Vα ) ⊂ Vα∗ ⊂ Vξ∗

β∈S0 ∩(γ,λ)

γ∈ξ∪{−1}

α∈S\ξ

α∈S\ξ

holds. It suffices (from Lemma 6.5 for the latter) to find a continuous shrinking {Fξ∗ : ξ ∈ S0 } of {Vξ∗ : ξ ∈ S0 }  by closed neighborhoods of q, q in Y × Y . Let Fζ∗ = ClY ×Y ( β∈S0 \ζ (Wβ × Wβ )) for each ζ ∈ S0 \ Lim(S0 ),  and let Fξ∗ = {Fζ∗ : ζ ∈ (S0 \ Lim(S0 )) ∩ ξ} for each ξ ∈ S0 ∩ Lim(S0 ). Then {Fξ∗ : ξ ∈ S0 } is a continuously descending sequence of closed neighborhoods of q, q in Y × Y . Let ξ ∈ S0 . Pick any y ∗ ∈ Fξ∗ . Take any γ ∈ ξ ∪ {−1}. Let ζ = min(S0 ∩ (γ, λ)) in case ξ ∈ S0 ∩ Lim(S0 ), otherwise put ζ = ξ. In any case, we have ζ ∈ S0 \ Lim(S0 ) and γ < ζ ≤ ξ. So it follows that y ∗ ∈ Fξ∗ ⊂ Fζ∗ = ClY ×Y (



(Wβ × Wβ )) ⊂ ClY ×Y (

β∈S0 \ζ





(Wβ × Wβ )).

β∈S0 ∩(γ,λ)



 ∗ ∗ ∗ Hence we obtain y ∗ ∈ γ∈ξ∪{−1} ClY ×Y β∈S0 ∩(γ,λ) (Wβ × Wβ ) ⊂ Vξ . This means that Fξ ⊂ Vξ holds ∗ for each ξ ∈ S0 . Thus {Fξ : ξ ∈ S0 } satisfies the required condition. 2 Normality of Δ-normal products Let F be a subset of a space Z. Recall that F is called a retract of Z if there is a continuous map f : Z → F such that f (z) = z for each z ∈ F . Note that every retract of a Hausdorff space is a closed set. Lemma 6.7. If F is a retract of a Δ-normal space Z, then F is normally located in Z. Proof. Let U be an open set in Z with F ⊂ U . Applying Δ-normality of Z for an open cover {U, Z \ F } of Z, it follows from Lemma 4.1 that there is an open cover W of Z such that ClZ×Z



   {W × W : W ∈ W} ⊂ (U × U ) ∪ (Z \ F ) × (Z \ F ) .

Take a continuous map f : Z → F such that f (z) = z for each z ∈ F . Then f −1 (W ) = {z ∈ Z : f (z) ∈ W }  is an open set in Z for each W ∈ W. Let V = {W ∩ f −1 (W ) : W ∈ W}. Then V is an open set in Z. Moreover, it is easily seen that F is contained in V . To see that ClZ (V ) ⊂ U , pick a z ∈ ClZ (V ). Let Q∗ be an open neighborhood of z ∗ = z, f (z) in Z × Z. Then there are neighborhoods Q0 , Q1 of z, f (z) in Z respectively such that Q0 × Q1 ⊂ Q∗ . By continuity,

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f −1 (Q1 ) = {z  ∈ Z : f (z  ) ∈ Q1 } is a neighborhood of z in Z. Since Q0 ∩ f −1 (Q1 ) is also a neighborhood of z in Z, there is a z  ∈ Q0 ∩ f −1 (Q1 ) ∩ V . And we can take a W0 ∈ W with z  ∈ W0 ∩ f −1 (W0 ). Then we have z  ∈ Q0 ∩ W0 and f (z  ) ∈ Q1 ∩ W0 , so z  , f (z  ) ∈ (Q0 × Q1 ) ∩ (W0 × W0 ) ⊂ Q∗ ∩ (W0 × W0 ). Hence  each neighborhood Q∗ of z, f (z) in Z × Z meets {W × W : W ∈ W}. Therefore 

z, f (z) ∈ ClZ×Z (

{W × W : W ∈ W}) ⊂ (U × U ) ∪ ((Z \ F ) × (Z \ F ))

holds. By f (z) ∈ F , we have z, f (z) ∈ / (Z \F ) ×(Z \F ), so z, f (z) ∈ U ×U , in particular z ∈ U holds. 2 Lemma 6.8. Let S be a stationary subset of a regular uncountable cardinal κ, and Y a space. If S × Y is Δ-normal, then Y has the weak S-nbdΔ property. Proof. Let V = {Vα : α ∈ S} be a collection of open neighborhoods of q ∈ Y . Put Sα = S ∩ [0, α] for each  α ∈ S. Then S × {q} ⊂ α∈S (Sα × Vα ). Since the function ξ, y → ξ, q witnesses that S × {q} is a retract of S × Y , by the above lemma, S × {q} is normally located in S × Y . Hence we can take an open set G∗ in  S × Y such that S × {q} ⊂ G∗ and ClS×Y G∗ ⊂ α∈κ (Sα × Vα ). Applying Δ-normality for an open cover {Sα × Vα : α ∈ κ} ∪ {(S × Y ) \ ClS×Y G∗ } of S × Y , there is an open cover U ∗ of S × Y such that  ∗    Cl(S×Y )×(S×Y ) {U × U ∗ : U ∗ ∈ U ∗ } ∩ ClS×Y G∗ × ClS×Y G∗    (Sα × Vα ) × (Sα × Vα ) . ⊂ α∈S

Since S × {q} is covered by a family {G∗ ∩ U ∗ : U ∗ ∈ U ∗ } of open sets in S × Y , by applying PDL, we can take an S  = S ∩ (γ0 , κ) for some γ0 ∈ κ such that: for each β ∈ S  , there are a neighborhood Wβ of q in Y and a Uβ∗ ∈ U ∗ such that Sβ × Wβ ⊂ G∗ ∩ Uβ∗ , where Sβ = S ∩ (γ0 , β] for each β ∈ S  . Notice that S  is of the form S  = S ∩ C for a club set C of κ by putting C = (γ0 , κ). To see that Y has the weak S-nbdΔ property at q, let ξ ∈ S  and y ∗ = y0 , y1  ∈

γ∈ξ∪{−1}

ClY ×Y





 (Wβ × Wβ ) .

β∈S  ∩(γ,κ)

 It suffices from Lemma 6.6(2) to show that y ∗ ∈ α∈S\ξ (Vα × Vα ). Let Q∗∗ be an open neighborhood of ξ, y0 , ξ, y1  in (S × Y ) × (S × Y ). Take a γ1 ∈ ξ ∪ {−1} with γ0 ≤ γ1 and an open neighborhoods Q0 and Q1 of y0 and y1 in Y , respectively, such that ((S ∩ (γ1 , ξ]) × Q0 ) × ((S ∩ (γ1 , ξ]) × Q1 ) ⊂ Q∗∗ . Take a ζ ∈ S ∩ (γ1 , ξ) and put γ = ζ in case ξ ∈ Lim(S). Put ζ = ξ and take a γ ∈ [γ1 , ξ) with S ∩ (γ, ξ) = ∅ / Lim(S). In any case, we have ζ ∈ S ∩ (γ1 , ξ] ⊂ S  , γ ∈ ξ ∪ {−1}, and S  ∩ (γ, κ) ⊂ S  ∩ [ζ, κ). in case ξ ∈   Since Q0 × Q1 is a neighborhood of y ∗ = y0 , y1  in Y × Y and y ∗ ∈ ClY ×Y β∈S  ∩(γ,κ) (Wβ × Wβ ) , we obtain a δ ∈ S  ∩ (γ, κ) ⊂ S  ∩ [ζ, κ) with (Q0 × Q1 ) ∩ (Wδ × Wδ ) = ∅. Take a point y0 , y1  of this non-empty set. Then Q∗∗ meets (Uδ∗ × Uδ∗ ) ∩ (G∗ × G∗ ). Actually, observe     ζ, y0 , ζ, y1  ∈ (S ∩ (γ1 , ξ]) × Q0 × (S ∩ (γ1 , ξ]) × Q1 ⊂ Q∗∗ .

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By ζ, δ ∈ S  with ζ ≤ δ, we have ζ ∈ Sδ , so ζ, y0 , ζ, y1  ∈ Sδ × Wδ ⊂ G∗ ∩ Uδ∗ . So we obtain ζ, y0 , ζ, y1  ∈ Q∗∗ ∩ (Uδ∗ × Uδ∗ ) ∩ (G∗ × G∗ ). This means that ξ, y0 , ξ, y1  ∈ Cl(S×Y )×(S×Y )





{U ∗ × U ∗ : U ∗ ∈ U ∗ }

∗

and

ξ, y0 , ξ, y1  ∈ Cl(S×Y )×(S×Y ) (G × G ) = ClS×Y G × ClS×Y G∗ . Thus we obtain that ξ, y0 , ξ, y1  ∈

 α∈S

∗

∗

  (Sα × Vα ) × (Sα × Vα ) . Take an α0 ∈ S with

ξ, y0 , ξ, y1  ∈ (Sα0 × Vα0 ) × (Sα0 × Vα0 ). Then we have ξ ≤ α0 and y0 , y1 ∈ Vα0 . Hence y ∗ = y0 , y1  ∈



α∈S\ξ (Vα

× Vα ) holds. 2

Since a space which is homeomorphic to a closed subspace of a Δ-normal space is also Δ-normal, the corollary below is obtained. Corollary 6.9. Let X and Y be spaces. If X × Y is Δ-normal, then • Y has the weak SE -nbdΔ property for each E ∈ S(X). • X has the weak SF -nbdΔ property for each F ∈ S(Y ). Thus the following is obtained as an immediate consequence of Corollary 6.9, Lemma 6.2(2) and Theorem 5.5. Corollary 6.10. Let X be a monotonically normal space, and K a compact space. If X × K is Δ-normal, then X × K is normal. 7. A characterization for Δ-paracompactness of products The four neighborhood properties on locally compact spaces The following facts for character are easily seen. Fact 7.1. Let κ be a regular cardinal. (1) If a space Y has the character less than κ at q ∈ Y , then Y has orthocaliber κ and the tightness < κ at q. (2) If two spaces X and Y have the character less than κ at p ∈ X and q ∈ Y respectively, then X × Y has the character less than κ at p, q. Moreover, the tightness < κ is preserved in the products with a locally compact factor. Fact 7.2. ([12]) Let κ be a regular cardinal. Let X and Y be spaces with p ∈ X and q ∈ Y . Assume that Y is locally compact. If X and Y have the tightness less than κ at p and q, respectively, then X × Y has the tightness less than κ. Let κ be a regular cardinal, S a set of ordinals and Y a space with q ∈ Y . Recall that Y has the κ-nbdΔ property (the weak S-nbdΔ property) at q if and only if Y × Y has the κ-dop property (S-docs property)

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at q, q. Then, what should we call the property that Y × Y has orthocaliber κ at q, q? However, we do not need a new name for this property. In fact, the following is easily seen. Lemma 7.3. Let κ be a regular cardinal. (1) If X and Y are spaces having orthocaliber κ at p ∈ X and q ∈ Y , respectively, then X × Y has orthocaliber κ at p, q. (2) Let Y be a space with q ∈ Y . Then Y × Y has orthocaliber κ at q, q if and only if Y has orthocaliber κ at q. On the other hand, for the κ-dop property and the S-docs property, we obtain some analogous results to Fact 7.2. Lemma 7.4. Let κ be a regular cardinal. Let X and Y be spaces with p ∈ X and q ∈ Y . Assume that Y is a locally compact space. (1) If X and Y has the κ-dop property at p and q, respectively, then X × Y has the κ-dop property at p, q. (2) Let S be an unbounded subset in κ. If X has the S-docs property at p and Y has the κ-dop property at q, then X × Y has the S-docs property at p, q. Proof. Let S be an unbounded subset in κ. Let {Vξ∗ : ξ ∈ S} be a descending sequence of open neighborhoods of p, q in X × Y . For each ξ ∈ κ \ S, we let Vξ∗ = Vξ∗+ , where ξ + = min(S \ ξ). Then {Vξ∗ : ξ ∈ κ} is also a descending sequence of open neighborhoods of p, q in X × Y (only consider as S = κ for (1)). Let Qξ = {y ∈ Y : p, y ∈ Vξ∗ } for each ξ ∈ κ. Then {Qξ : ξ ∈ κ} is a descending sequence of open neighborhoods of q in Y . Since Y is a locally compact space with the κ-dop property, we can take an open neighborhood  Q of q in Y with ClY Q ⊂ ξ∈κ Qξ such that ClY Q is compact. Let Pξ = {x ∈ X : {x} × ClY Q ⊂ Vξ∗ } for each ξ ∈ κ. Then {Pξ : ξ ∈ κ} is a descending sequence of open neighborhoods of p in X since {p} × ClY Q ⊂ {p} × Qξ ⊂ Vξ∗ for each ξ ∈ κ.  (1): By the assumption of X, we obtain an open neighborhood P of p in X with P ⊂ ξ∈κ Pξ . Then we have P × Q ⊂ Pξ × Q ⊂ Vξ∗ for each ξ ∈ κ, so W ∗ = P × Q is an open neighborhood of p, q with  W ∗ ⊂ ξ∈κ Vξ∗ . (2): By the assumption of X, we obtain a continuously descending sequence {Fξ : ξ ∈ S} of closed neighborhoods of p in X such that Fξ ⊂ Pξ for each ξ ∈ S. Then {Fξ × ClY Q : ξ ∈ S} is also a continuously descending sequence of closed neighborhoods of p, q in X ×Y such that Fξ ×ClY Q ⊂ Vξ∗ for each ξ ∈ S. 2 Thus, the orthocaliber κ property and the κ-dop property behave like the properties having < κ-character and < κ-tightness, respectively. As a consequence, the following result is obtained. Corollary 7.5. Let S be a stationary subset of a regular uncountable cardinal κ. Let Y be a locally compact space with q ∈ Y . Then the following are equivalent: (a) (b) (c) (d)

Y Y Y Y

has has has has

the the the the

κ-nbdΔ-property at q. κ-dop property at q. weak S-nbdΔ-property at q. S-docs property at q.

Proof. It follows from the diagram of implications described after Lemma 6.2 that (a) ⇒ (b) ⇒ (d) and (a) ⇒ (c) ⇒ (d) hold. For a locally compact space Y , (d) ⇒ (b) was proved in [9, Lemma 4.2] and (b) ⇒ (a) is assured by Lemmas 5.2 and 7.4(1). 2

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Δ-paracompactness of products Although the following lemma will be used in the last section, we prove it here for the next lemma. Lemma 7.6. Let X be an orthocompact space, and Y a space. Let E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X). Assume that Y has the κ-nbdΔ property at q ∈ Y . Then for each family G of open sets of X × Y with    E × {q} ⊂ G, there is an open rectangle P × Q in X × Y with eE SE ∩ (γ, κ) ⊂ P for some γ ∈ κ and q ∈ Q such that: for each z0 , z1 ∈ P × Q, there is a G ∈ G with z0 , z1 ∈ G. Proof. Let e = eE and S = SE . Let U be the family of all open sets U in X such that U × V is contained in some member of G for some open neighborhood V of q in Y . Then U covers E. By Lemma 3.3 and Remark 3.4,    there is an ascending sequence {Pα : α ∈ κ} by members of U such that e S ∩ (γ, κ) ⊂ α∈κ Pα holds for  some γ ∈ κ. Let P = α∈κ Pα . For each α ∈ κ, by Pα ∈ U, we can take an open neighborhood Vα of q in Y and a Gα ∈ G with Pα × Vα ⊂ Gα . By applying the κ-nbdΔ property for {Vα : α ∈ κ}, we can take an open neighborhood Q of q in Y such that: for each y0 , y1 ∈ Q, there are unboundedly many α ∈ κ with y0 , y1 ∈ Vα . Let z0 = x0 , y0 , z1 = x1 , y1  ∈ P × Q. Then there are α0 , α1 ∈ κ with x0 ∈ Pα0 and x1 ∈ Pα1 . And there is an α2 ∈ κ with α0 , α1 ≤ α2 such that y0 , y1 ∈ Vα2 . We have x0 ∈ Pα0 ⊂ Pα2 by α0 ≤ α2 , and x1 ∈ Pα1 ⊂ Pα2 by α1 ≤ α2 . Therefore z0 , z1 ∈ Pα2 × Vα2 ⊂ Gα2 holds with Gα2 ∈ G. Hence P × Q is a required open rectangle in X × Y . 2 Applying Lemma 7.6 for X = E, we immediately obtain a special case of it, which is necessary for the purpose in this section. Lemma 7.7. Let X and Y be spaces. Let E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X). Assume that Y has the  κ-nbdΔ property at q ∈ Y . Then for each family G of open sets of X × Y with E × {q} ⊂ G, there are a   γ ∈ κ and an open neighborhood Q of q in Y such that: for each z0 , z1 ∈ eE SE ∩ (γ, κ) × Q, there is a G ∈ G with z0 , z1 ∈ G. Lemma 7.8. Let X and Y be spaces. Let E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X). Assume that Y is regular and has the κ-nbdΔ property at q ∈ Y . And assume that the projection π ∗ : (X × X) × (Y × Y ) → X × X  is a closed map. Then for each family G of open sets in X × Y with E × {q} ⊂ G, there are a γ ∈ κ, a family U of open sets in X and an open neighborhood Q of q in Y , satisfying that  (i) eE (SE ∩ (γ, κ)) ⊂ U, (ii) if U ∈ U and z0 , z1 ∈ U × Q, then there is a G ∈ G with z0 , z1 ∈ G. Proof. By Lemma 7.7, there are a γ ∈ κ and an open neighborhood W of q in Y such that: for each     z0 , z1 ∈ eE SE ∩ (γ, κ) × W , there is a G ∈ G with z0 , z1 ∈ G. Let A = eE SE ∩ (γ, κ) . Then A is closed in X. Let X ∗ = X × X, Y ∗ = Y × Y , q ∗ = q, q, A∗ = A × A, B ∗ = {q ∗ }, W ∗ = W × W . For each G ∈ G, let O∗ (G) = {x0 , x1 , y0 , y1  ∈ X ∗ × Y ∗ : x0 , y0 , x1 , y1  ∈ G},  and let O∗ = G∈G O∗ (G). Then A∗ ⊂ X ∗ , q ∗ ∈ B ∗ ⊂ W ∗ ⊂ Y ∗ , B ∗ is normally located in Y ∗ , W ∗ is open in Y ∗ and O∗ is an open set in X ∗ × Y ∗ with A∗ × B ∗ ⊂ A∗ × W ∗ ⊂ O∗ . Since the projection π ∗ : X ∗ × Y ∗ → X ∗ is a closed map, it follows from Lemma 3.8 that there is an open rectangle P ∗ × Q∗ in X ∗ × Y ∗ such that A × A = A∗ ⊂ P ∗ , q, q = q ∗ ∈ B ∗ ⊂ Q∗ and P ∗ × Q∗ ⊂ O∗ . Let U = {U ⊂ X : U is open in X with U × U ⊂ P ∗ }

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and let Q be an open neighborhood of q in Y with Q × Q ⊂ Q∗ . Then (i) is obviously satisfied. To show that (ii) is satisfied, pick any z0 = x0 , y0 , z1 = x1 , y1  ∈ U × Q, where U ∈ U. Since x0 , x1 , y0 , y1  ∈ (U × U ) × (Q × Q) ⊂ P ∗ × Q∗ ⊂ O∗ , there is a G0 ∈ G with x0 , x1 , y0 , y1  ∈ O∗ (G0 ). This means z0 = x0 , y0 , z1 = x1 , y1  ∈ G0 . 2 Now, we give a main theorem in this section as follows. Theorem 7.9. Let X be a monotonically normal space and K a non-empty compact space. Then X × K is Δ-paracompact if and only if X is Δ-paracompact and X × K is normal. Proof. The “only if” part immediately follows from Corollary 5.6. Assume that X is Δ-paracompact and X × K is normal. By Theorem 5.5, K has the κ-dop property for each κ ∈ S ∗ (X). Moreover, it follows from Corollary 7.5 that K has the κ-nbdΔ property for each κ ∈ S ∗ (X). We show that X × K is Δ-paracompact. Let G be an open cover of X × K. Let H be the family of all open sets H in X × K such that: for each z0 , z1 ∈ H, there is a G ∈ G with z0 , z1 ∈ G. Then H is an open cover of X × K since H contains G. It suffices to show that H has a locally finite subcover (open refinement). By Lemma 2.3 ([17, Lemma 8.2]), it suffices to show that: for each E ∈ S(X, κ) with κ ∈ S ∗ (X) and for each q ∈ K, there is an open rectangle   P × Q in X × K such that eE SE ∩ (γ, κ) ⊂ P for some γ ∈ κ, q ∈ Q, and P × Q is contained in some member of H. For such a κ, K has the κ-nbdΔ-property at q. Since K × K is compact, the projection π ∗ : (X × X) × (K × K) → X × X is a closed map. By Lemma 7.8, there are a γ0 ∈ κ, a family U of open sets in X and an open neighborhood Q of q in Y , satisfying that  (i) eE (SE ∩ (γ0 , κ)) ⊂ U, (ii) if U ∈ U and z0 , z1 ∈ U × Q, then there is a G ∈ G with z0 , z1 ∈ G. Since X is Δ-paracompact, by Lemma 3.2, there is an open set P in X with eE (SE ∩ (γ, κ)) ⊂ P for some γ ∈ κ such that: for each x0 , x1 ∈ P , there is a U ∈ U with x0 , x1 ∈ U . We obtain an open rectangle P × Q in X × K. To see that P × Q is a required one, it suffices to show that P × Q ∈ H. Pick any z0 = x0 , y0 , z1 = x1 , y1  ∈ P × Q. By x0 , x1 ∈ P , there is a U ∈ U with x0 , x1 ∈ U . Since z0 , z1 ∈ U × Q, we found by (ii) a G ∈ G with z0 , z1 ∈ G. Hence P × Q ∈ H holds. Therefore X × K is Δ-paracompact. 2 8. A characterization for Δ-normality of products Since Δ-paracompact and normal spaces are Δ-normal in general [3,13], the following is an immediate consequence of Corollary 6.10 and Theorem 7.9. Corollary 8.1. Let X be a Δ-paracompact and monotonically normal space and K a compact space. Then the following are equivalent: (a) X × K is Δ-paracompact. (b) X × K is Δ-normal. (c) X × K is normal. In this section, we show that: for deriving the equivalence (b) ↔ (c), Δ-paracompactness of X in the assumption can be replaced by Δ-normality.

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Lemma 8.2. Let Z ∗ be a regular space. For each family G ∗ of open sets in Z ∗ , let H(G ∗ ) be the class of all families H∗ of open sets in Z ∗ such that: ClZ ∗ ×Z ∗

  {H ∗ × H ∗ : H ∗ ∈ H∗ } ⊂ {G∗ × G∗ : G∗ ∈ G ∗ }.



 (1) For each open cover G ∗ of Z ∗ , { H∗ : H∗ ∈ H(G ∗ )} is an open cover of Z ∗ .  (2) If { H∗ : H∗ ∈ H(G ∗ )} has a locally finite open refinement in Z ∗ for each open cover G ∗ of Z ∗ , then Z ∗ is Δ-normal. Proof. (1) Pick a z ∗ ∈ Z ∗ . Take a G∗0 ∈ G ∗ with z ∗ ∈ G∗0 . And take an open neighborhood H0∗ of z ∗ in Z ∗ with ClZ ∗ H0∗ ⊂ G∗0 . Let H∗ = {H0∗ }. Then we have 

 {H ∗ × H ∗ : H ∗ ∈ H∗ } = ClZ ∗ ×Z ∗ (H0∗ × H0∗ )  ⊂ G∗0 × G∗0 ⊂ {G∗ × G∗ : G∗ ∈ G ∗ }.

ClZ ∗ ×Z ∗

  So H∗ ∈ H(G) with z ∗ ∈ H0∗ = H∗ . Hence { H∗ : H∗ ∈ H(G ∗ )} is an open cover of Z ∗ . (2) Let G ∗ be an open cover of Z ∗ . To see that Z ∗ is Δ-normal, it suffices to find an open cover U ∗ of Z ∗  with U ∗ ∈ H(G ∗ ). By the assumption, we obtain a locally finite open refinement L∗ of { H∗ : H∗ ∈ H(G ∗ )}.  For each L∗ ∈ L∗ , take and fix an H∗ (L∗ ) ∈ H(G ∗ ) with L∗ ⊂ H∗ (L∗ ). It is routine to check that U ∗ = {L∗ ∩ H ∗ : L∗ ∈ L∗ and H ∗ ∈ H∗ (L∗ )} is a required open cover of Z ∗ . 2 Theorem 8.3. Let X be a monotonically normal space and K a non-empty compact space. Then X × K is Δ-normal if and only if X is Δ-normal and X × K is normal. Proof. The “only if” part immediately follows from Corollary 6.10. Assume that X is Δ-normal and X × K is normal. By Theorem 5.5 and Corollary 7.5, note that K has the κ-nbdΔ property for each κ ∈ S ∗ (X). Notice that the projection π ∗ : (X × X) × (K × K) → X × X, x0 , x1 , y0 , y1  → x0 , x1 , is a closed map. Now, we prove that Z ∗ := X × K is Δ-normal. Let G ∗ be an open cover of Z ∗ . Define H(G ∗ ) as in Lemma 8.2. Take an E ∈ S(X, κ) with κ ∈ S ∗ (X) and pick a q ∈ K. It suffices from Lemmas 8.2(2) and 2.3 to find an open rectangle P × Q in Z ∗ with  eE (SE ∩ (γ, κ)) ⊂ P for some γ ∈ κ and q ∈ Q, and an H∗ ∈ H(G ∗ ) such that P × Q ⊂ H∗ . By  Lemma 7.8, we can take a γ0 ∈ κ, a family U of open sets in X with eE (SE ∩ (γ0 , κ)) ⊂ U, and an open neighborhood Q of q in Y such that: for each z0∗ , z1∗ ∈ U × Q with U ∈ U, there is a G∗ ∈ G ∗ with z0∗ , z1∗ ∈ G∗ . Since X is monotonically normal and Δ-normal, it follows from Lemma 4.2 that there is a  family V of open sets in X with eE (SE ∩ (γ, κ)) ⊂ V for some γ ∈ κ such that ClX×X



  {V × V : V ∈ V} ⊂ {U × U : U ∈ U}.

 Let P = V. Take an open neighborhood Q of q in Y with ClY Q ⊂ Q . And let H∗ = {V × Q : V ∈ V}.  Then H∗ is a family of open sets in Z ∗ , P × Q is an open rectangle in X × Y and P × Q = H∗ . To see that P × Q and H∗ satisfies the required conditions, it suffices to show that H∗ ∈ H(G ∗ ), i.e. ClZ ∗ ×Z ∗



  {H ∗ × H ∗ : H ∗ ∈ H∗ } ⊂ {G∗ × G∗ : G∗ ∈ G ∗ }

 ∗  holds. Pick a z0∗ , z1∗  ∈ ClZ ∗ ×Z ∗ {H × H ∗ : H ∗ ∈ H∗ } with z0∗ = x0 , y0  and z1∗ = x1 , y1 . Then    we have x0 , x1  ∈ ClX×X {V × V : V ∈ V} ⊂ {U × U : U ∈ U} and y0 , y1 ∈ ClY Q ⊂ Q . So z0∗ , z1∗ ∈ U0 × Q holds for some U0 ∈ U, hence there is a G∗0 ∈ G ∗ with z0∗ , z1∗ ∈ G∗0 . Thus we conclude that  z0∗ , z1∗  ∈ {G∗ × G∗ : G∗ ∈ G ∗ }. 2

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9. Paracompact DC DC-like spaces Let DC denote the class of all spaces which have a discrete cover by compact sets. Definition 9. A space X is said to be DC-like [4,15] if for each closed set F in X, one can assign a closed set s(F ) ∈ DC with s(F ) ⊂ F , satisfying that: if {Fn } is a descending sequence of closed sets in X such that s(Fn ) ∩ Fn+1 = ∅ for each n ∈ ω, then  n∈ω Fn = ∅ holds. Every compact space is obviously DC-like. It is known that every space with a closure-preserving cover by compact sets or every subparacompact C-scattered space is DC-like (see [15]). Lemma 2.3 was generalized as below. Lemma 9.1. ([17, Lemmas 8.2 and 8.3]) Let X be a monotonically normal space and Y a paracompact DC-like space. Let G be an open cover of X × Y , satisfying that for each E = eE (SE ) ∈ S(X, κ) with κ ∈ S ∗ (X) and for each q ∈ Y , there is an open rectangle P × Q in X × Y such that eE (SE ∩ (γ, κ)) ⊂ P for some γ ∈ κ, q ∈ Q and P × Q is contained in some member of G. Then G has a σ-locally finite rectangular cozero refinement. In particular, G is a normal cover of X × Y . Theorem 9.2. Let X be a monotonically normal and orthocompact space and Y a paracompact DC-like space. Then X × Y is Δ-paracompact if and only if Y has the κ-nbdΔ property for each κ ∈ S ∗ (X). Proof. The “only if” part immediately follows from Corollary 5.4. Assume that Y has the κ-nbdΔ property for each κ ∈ S ∗ (X). Let G be an open cover of X × Y . Let H be the family of all open sets H in X × Y such that: for each z0 , z1 ∈ H, there is a G ∈ G with z0 , z1 ∈ G. Then H is an open cover of X × Y since H contains G. It suffices to show that H has a locally finite subcover (open refinement). Moreover, it suffices from Lemma 9.1 to show that: for each E ∈ S(X, κ) with κ ∈ S ∗ (X) and for each q ∈ Y , there is an open   rectangle P × Q in X × Y such that eE SE ∩ (γ, κ) ⊂ P for some γ ∈ κ, q ∈ Q, and P × Q is contained in some member of H. For such κ, it is assumed that Y has the κ-nbdΔ-property. Therefore Lemma 7.6 assures that such a rectangle P × Q exists as a member of H. Hence X × Y is Δ-paracompact. 2 Lemma 9.3. ([11]) Let S be a stationary subset of a regular uncountable cardinal κ, and Y a space. If S × Y is orthocompact, then Y has orthocaliber κ. Corollary 9.4. Let X be a monotonically normal space and Y a paracompact DC-like space. If X × Y is orthocompact, then X × Y is Δ-paracompact. Proof. Assume that X ×Y is orthocompact and Y is non-empty. Take a κ ∈ S ∗ (X). Then X is orthocompact and Y has orthocaliber κ by Lemma 9.3, and so it has the κ-nbdΔ property by Lemma 5.1(1). Therefore X × Y is Δ-paracompact by Theorem 9.2. 2 Lemma 9.5. ([9, Theorem 5.2]) Let X be a monotonically normal and orthocompact space and Y a paracompact DC-like space. Then X × Y is normal and rectangular if and only if Y has the κ-dop property for each κ ∈ S ∗ (X). Corollary 9.6. Let X be a monotonically normal and orthocompact space and Y a paracompact DC-like space. If X × Y is Δ-paracompact, then X × Y is normal and rectangular.

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Proof. If X × Y is Δ-paracompact, then Y has the κ-nbdΔ property by Corollary 5.4, and so it has the κ-dop property for each κ ∈ S ∗ (X) by Lemma 5.1(2). Hence it follows from Lemma 9.5 that X × Y is normal and rectangular. 2 Lemma 9.7. Let X be a monotonically normal space and Y a paracompact DC-like space. For each open cover G ∗ of X × Y , let H(G ∗ ) be as described in Lemma 8.2. Assume that for each open cover G ∗ of X × Y , for each E ∈ S(X, κ) with κ ∈ S ∗ (X) and for each q ∈ Y , there are an open rectangle P × Q in X × Y  with eE (SE ∩ (γ, κ)) ⊂ P for some γ ∈ κ and q ∈ Q, and an H∗ ∈ H(G ∗ ) with P × Q ⊂ H∗ . Then Z ∗ is Δ-normal.  Proof. By the assumption, it follows from Lemma 9.1 that { H∗ : H∗ ∈ H(G ∗ )} is a normal cover of X ×Y . So this immediately follows from Lemma 8.2(2). 2 Lemma 9.8. Let X be a monotonically normal and orthocompact space and Y a space. Let E ∈ S(X, κ) with κ ∈ S ∗ (X) and assume that Y has the weak SE -nbdΔ property at q ∈ Y . Then for each family G ∗ of open sets     in X ×Y with E ×{q} ⊂ G ∗ , there is a family H∗ of open sets in X ×Y with eE SE ∩(γ, κ) ×{q} ⊂ H∗ for some γ ∈ κ such that: Cl(X×Y )×(X×Y )



  {H ∗ × H ∗ : H ∗ ∈ H∗ } ⊂ {G∗ × G∗ : G∗ ∈ G ∗ }.

Proof. Let e = eE and S = SE . Let P be the family of all open sets P in X such that P × V is contained in some member of G ∗ for some open neighborhood V of q in Y . Then P covers E. Using Lemma 3.3, we    can take an ascending sequence {Pα : α ∈ κ} by members of P such that e S ∩ (γ, κ) ⊂ α∈κ Pα holds    for some γ ∈ κ. Let P = α∈κ Pα . Let Eα = ∅ for each α ≤ γ, and Eα = e S ∩ (γ, α] for each α ∈ (γ, κ).  Replacing the index, we may assume that Eα ⊂ Pα for each α ∈ κ, and Pα = β∈α Pβ for each limit ordinal α ∈ κ \ S. Since there is a monotonically normal operator for X, we can take an ascending sequence {Pα : α ∈ κ} of open sets in X such that Eα ⊂ Pα ⊂ ClX Pα ⊂ Pα for each α ∈ κ. Since X is normal and    Eα ⊂ e S ∩ (γ, κ) ⊂ P for each α ∈ κ, we may assume that ClX P  ⊂ P , where P  = α∈κ Pα . For each α ∈ κ, take an open neighborhood Vα of q in Y , such that Pα × Vα ⊂ G∗α for some G∗α ∈ G ∗ . Applying the weak S-nbdΔ property, it follows from Lemma 6.6 that there is a family {Qβ : β ∈ S} of open neighborhoods of q in Y such that: for each ξ ∈ S,

ClY ×Y

γ∈ξ∪{−1}





  (Qβ × Qβ ) ⊂ (Vα × Vα ).

β∈S∩(γ,κ)

α∈S\ξ

   Put H∗ = {Pβ ×Qβ : β ∈ S}. Obviously, H∗ is a family of open sets in X ×Y with e S ∩(γ, κ) ×{q} ⊂ H∗ . To see that H∗ is a required one, let z0∗ , z1∗  ∈ Cl(X×Y )×(X×Y )



 {H ∗ × H ∗ : H ∗ ∈ H∗ } .

 It suffices to show that z0∗ , z1∗  ∈ {G∗ × G∗ : G∗ ∈ G ∗ }. Let zi∗ = xi , yi  for each i ∈ 2. By zi∗ ∈  ∗  ∗ ClX×Y ( H ) and H ⊂ P  × Y , we have xi ∈ ClX P  ⊂ P , so there is a ξ ∈ S with x0 , x1 ∈ Pξ . Take   the least such ξ. Let γ ∈ ξ ∪ {−1}. We show that xi ∈ / ClX {Pβ : β ∈ S ∩ [0, γ]} for some i ∈ 2. We may  let S ∩ [0, γ] = ∅. In case S ∩ [0, γ] ⊂ [0, ζ] for some ζ ∈ S ∩ ξ, we have {Pβ : β ∈ S ∩ [0, γ]} ⊂ Pζ , and so ClX



 {Pβ : β ∈ S ∩ [0, γ]} ⊂ ClX Pζ ⊂ Pζ  / xi

holds for some i ∈ 2 by the minimality of ξ. In the other case, S∩[0, γ] is non-empty and does not have a max imum. Then ζ := sup(S ∩ [0, γ]) is a limit ordinal with ζ ∈ κ \ S. It follows from {Pβ : β ∈ S ∩ [0, γ]} ⊂ Pζ

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that ClX



   {Pβ : β ∈ S ∩ [0, γ]} ⊂ ClX Pζ ⊂ Pζ = Pβ = Pβ  / xi β∈ζ

β∈S∩ζ

for some i ∈ 2 by the minimality of ξ. We see that xi ∈ / ClX ({Pβ : β ∈ S ∩ [0, γ]}) for some i ∈ 2 in any case. Therefore it follows that   z0∗ , z1∗  ∈ Cl(X×Y )×(X×Y ) {(Pβ × Qβ ) × (Pβ × Qβ ) : β ∈ S ∩ (γ, κ)} , and so y0 , y1  ∈ ClY ×Y



β∈S∩(γ,κ) (Qβ

y0 , y1  ∈

γ∈ξ∪{−1}

 × Qβ ) holds. We have

ClY ×Y



 β∈S∩(γ,κ)

  (Qβ × Qβ ) ⊂ (Vα × Vα ). α∈S\ξ

By taking α ∈ S \ ξ with y0 , y1  ∈ Vα × Vα , we have x0 , x1 ∈ Pξ ⊂ Pα . Hence zi∗ = xi , yi  ∈ Pα × Vα ⊂ G∗α  holds for each i ∈ 2. We found a G∗α ∈ G ∗ with z0∗ , z1∗  ∈ G∗α × G∗α , so z0∗ , z1∗  ∈ {G∗ × G∗ : G∗ ∈ G ∗ } holds. 2 Theorem 9.9. Let X be a monotonically normal and orthocompact space and Y a paracompact DC-like space. Assume that X × Y is normal and rectangular. Then X × Y is Δ-normal if and only if Y has the weak SE -nbdΔ property for each E ∈ S(X). Proof. The “only if” part immediately follows from Corollary 6.9. To show the “if” part, assume that Y has the weak SE -nbdΔ property for each E ∈ S(X). Let G ∗ be an open cover of X × Y . Take an E ∈ S(X, κ) with κ ∈ S ∗ (X) and q ∈ Y . It follows from Lemma 9.8 that there is a family H∗ of open sets in X × Y    belonging to H(G ∗ ) such that eE SE ∩ (γ0 , κ) × {q} ⊂ H∗ for some γ0 ∈ κ. Since X × Y is normal and rectangular, it is easily seen that there are an open rectangle P × Q in X × Y and a γ ∈ κ with γ0 ≤ γ such    that eE SE ∩ (γ, κ)] × {q} ⊂ P × Q ⊂ H∗ . By Lemma 9.7, X × Y is Δ-normal. 2 It is necessary to consider the opposite implications in the diagram after Lemma 6.2. Several examples for them will be discussed in the next paper [8]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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