grabnerCWN.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 5, No. 2, 2004 pp. 199-212 Relative Collectionwise Normality Elise Grabner, Gary Grabner, Kazumi Miyazaki and Jamal Tartir Abstract. In this paper we study properties of relative collec- tionwise normality type based on relative properties of normality type introduced by Arhangel’skii and Genedi. Theorem Suppose Y is strongly regular in the space X. If Y is para- compact in X then Y is collectionwise normal in X. Example A T2 space X having a subspace which is 1− paracompact in X but not collectionwise normal in X. Theorem Suppose that Y is s- regular in the space X. If Y is meta- compact in X and strongly collectionwise normal in X then Y is para- compact in X. 2000 AMS Classification: Primary 54D20, Secondary 54A35 Keywords: paracompact, collectionwise normal, relative topological proper- ties 1. Introduction In this paper properties of relative collectionwise normality type based on relative properties of normality type introduced in [2] and [3] are studied. Our study focusses on the following well known theorems and relative properties of paracompactness type introduced in [1] and [4]. Theorem 1.1 (Bing). Every paracompact space is collectionwise normal. Theorem 1.2 (Michael-Nagami). Every metacompact collectionwise normal space is paracompact. A theorem concerning the relative properties of a subspace Y in a space X becomes a theorem about the corresponding global properties of X by let- ting Y = X. It is not surprising when the proof of a result concerning relative properties is a straight forward modification of the usual proof of the corre- sponding global result. For example we show that if Y is strongly star normal in X then Y is strongly collectionwise normal in X, Theorem 3.7. The proof is 200 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir the natural relative version of the standard proof that T2 paracompact spaces are collectionwise normal using the fully normal characterization of paracom- pactness. However this is not always true. For example there exist a good number of non-equivalent relative properties of paracompactness type, see [1], [2], [6], [7] and [8]. Some of these properties are preserved by closed maps (cp-paracompact in X, [7]) and some are not (paracompact in X from outside, [7]). Some imply that the subspace Y is paracompact (strongly star normal in X, [4]) while others do not (1- paracompact in X, [6]). We give an exam- ple of a T2 space having a subspace which is 1− paracompact in X but not collectionwise normal in X, Example 5.4. Thus to obtain an analog of Bing′s Theorem for subspaces Y paracompact in X it is necessary to assume that Y satisfies relative separation properties not implied by the space X being a T2 space and Y being paracompact in X. If Y is paracompact in X and strongly regular in X then Y is collectionwise normal in X, Theorem 3.3. We give several relative versions of the Michael-Nagami Theorem. If Y is s− regular in X, metacompact in X and strongly collectionwise normal in X then Y is paracompact in X, Theorem 4.4. If Y is closed, s− regular in X, collectionwise normal in X and metacompact then Y is paracompact in X, Corollary 4.5. Throughout this paper all spaces are assumed to be Hausdorff. Suppose X is a space and Y a subspace of X. When a set U is said to be open, we mean open with respect to the topology on X even if U happens to be a subset of Y. For a set X, x ∈ X, a subset A of X and a collection U of subsets of X, (U)x = {U ∈ U : x ∈ U}, (U)A = {U ∈ U : A ∩ U 6= φ}, st(x, U) = ∪(U)x and st(A, U) = ∪(U)A. 2. Definitions and Lemma Suppose Y is a subset of the space X. The subset Y is 1. regular in X , 2. super regular in X, 3. strongly regular in X, 4. s- regular in X, 5. normal in X , 6. s- normal in X, 7. strongly normal in X provided 1. for each x ∈ Y and every subset F of X\{x} closed in X there are disjoint open sets U and V such that x ∈ U and F ∩ Y ⊆ V [3]. 2. for each x ∈ Y and every subset F of X\{x} closed in X there are disjoint open sets U and V such that x ∈ U and F ⊆ V [3]. 3. for each x ∈ X and every subset F of X\{x} closed in X there are disjoint open sets U and V such that x ∈ U and F ∩ Y ⊆ V [3]. 4. Y is both super regular and strongly regular in X. 5. for each pair E and F of disjoint closed subsets of X there are disjoint open sets U and V such that E ∩ Y ⊆ U and F ∩ Y ⊆ V [3]. 6. for each pair, E and F of disjoint closed subsets of X, there are disjoint open subsets of X, U and V such that E ⊆ U and F ∩ Y ⊆ V [10]. 7. for each pair E and F of disjoint closed (in Y ) subsets of Y there are disjoint open sets U and V such that E ⊆ U andF ⊆ V [2]. Relative Collectionwise Normality 201 Suppose Y is a subset of a space X. If Y is super regular or strongly regular in X (s− normal or strongly normal in X) then Y is regular (normal) in X. However in general there is no implication between these two stronger condi- tions. Also if Y is normal (s- normal) in X then Y is regular (s- regular) in X. If X is a regular (normal) space then every subspace of X is s− regular (s- normal but not necessarily strongly normal) in X. The subspace Y can be strongly normal in X without being strongly regular in X. Suppose Y is a subset of a space X. A collection U is said to be locally finite on Y provided for every y ∈ Y there is an open V containing y such that (U)V is finite. A collection F of closed subsets of X is said to be weakly closure reserving with respect to Y provided for all F′ ⊆ (F) Y , (∪F′)∩Y = (∪F′)∩Y, [7]. The following lemmas from [7] are frequently used when working with collections that are locally finite with respect to a subset Y of a space X. Lemma 2.1. Suppose Y ⊆ X and U is a collection of open subsets of the space X locally finite on Y. Then the collection {U : U ∈ U} is weakly closure preserving with respect to Y and locally finite on Y. Lemma 2.2. Suppose that Y ⊆ X and F is a collection of closed subsets of the space X weakly closure preserving with respect to Y. 1. If B ⊆ X is closed then {F ∩ B : F ∈ F} is weakly closure preserving with respect to Y. 2. If A ⊆ Y then A ⊆ X\∪(F\(F)A). In particular, for all y ∈ Y, y /∈ ∪{F ∈ F : y /∈ F}. For a space X and Y ⊆ X, a collection A of subsets of the space X is said to be discrete with respect to Y provided for all x ∈ Y there is an open neighborhood U of x that intersects at most one member of A. We say that Y is collectionwise normal in a space X provided for every discrete collection F of closed subsets of X, there is a collection of open subsets of X, U = {U(F) : F ∈ F} discrete with respect to Y such that for all F ∈ F, F ∩ Y ⊆ U(F) ⊆ X\∪(F\{F}). Notice that a collection of subsets of a space X which is discrete with respect to a subspace Y of X need not be pairwise disjoint. However in the case of collectionwise normality in X this is not a problem as seen in the following lemma. Lemma 2.3. Suppose Y ⊆ X and U is a collection of open subsets of the space X discrete with respect to Y. For each U ∈ U let V (U) = U\∪(U\{U}). Then the collection {V (U) : U ∈ U} is a pairwise disjoint collection of open subsets of X discrete with respect to Y such that for all U ∈ U, U ∩ Y = V (U) ∩ Y. Theorem 2.4. If Y is collectionwise normal in the space X then Y is normal in X. We say that a subspace Y is strongly collectionwise normal in the space X provided for every collection F of closed subsets of X which is discrete with respect to Y there is a collection of open subsets of X, U = {U(F) : F ∈ F} discrete with respect to Y such that for all F ∈ F, F ∩ Y ⊆ U(F) ⊆ 202 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir X\∪(F\{F}). By Lemma 2.3 the members of U can be taken to be pairwise disjoint and discrete with respect to Y if we choose. Notice that if Y is a closed subset of X and F is a collection of closed subsets of X which is discrete with respect to Y then {F ∩ Y : F ∈ F} is a discrete collection of closed subsets of X. Theorem 2.5. If Y is strongly collectionwise normal in the space X then Y is strongly normal in X and a collectionwise normal subspace of X. If Y is a closed subset of X then Y is strongly collectionwise normal in X if and only if Y is collectionwise normal in X. A closed collectionwise normal subspace of a space X need not be collec- tionwise normal in X, Example 5.2. 3. Relative paracompact implies relative collectionwise normality The following definitions of the most natural properties of relative paracom- pactness type are from [2]. The subspace Y is said to be 1− paracompact in X provided every open cover of X has an open refinement locally finite on Y. The subspace Y is paracompact in X provided every open cover of X has an open partial refinement covering Y and locally finite on Y. In [6] it is observed that if Y is strongly regular in X and paracompact in X then Y is normal in X. If Y is closed and paracompact in X then Y is normal in X. However a closed subset of a regular space X can be paracompact in X and not s− normal in X, Example 5.3. Although it is readily seen that if Y is 1− paracompact in X then Y is super- regular in X it need not be strongly regular in X, Example 5.1. The following Theorem shows that s− normality in X is a relative property of normality type that relates to 1− paracompactness in X. Theorem 3.1. Suppose Y is strongly regular in the space X. If Y is 1− para- compact in X then Y is s−normal in X. Proof. Suppose E and F are disjoint closed subsets of X. Since Y is strongly regular in X, for every x ∈ E there are disjoint open sets W(x) and G(x) such that x ∈ W(x) and F ∩Y ⊆ G(x). Let W = {W(x) : x ∈ E}∪{X\E} and V be and open refinement of W locally finite on Y. For each V ∈ (V)E let x(V ) ∈ E such that V ⊆ W(x(V )). Let U = ∪(V)E and note that since V is a cover of X, E ⊆ U. Let O = X\U. Suppose x ∈ F ∩Y. Since V is locally finite on Y, let Q be an open neighborhood of x meeting only finitely many members of V. Let V′ = {V ∈ (V) E : Q ∩ V 6= φ} and note that V′ is finite. If V′ = φ then Q ∩ U = φ and so x ∈ O. Suppose V′ 6= φ, say V′ = {V1, V2, .., Vn}. Then Q ∩ G(x(V1)) ∩ ... ∩ G(x(Vn)) is an open neighborhood of x missing U and so again x ∈ O. Therefore F ∩ Y ⊆ O. � A space X can have a subspace which is 1− paracompact in X but not collectionwise normal in X, Example 5.4.This example is not regular and the subspace Y is not closed. Relative Collectionwise Normality 203 Theorem 3.2. Suppose that Y is closed and paracompact in the space X. Then Y is strongly collectionwise normal in X. Proof. By Theorem 2.5 we need only show that Y is collectionwise normal in X. Let {Fα : α ∈ Γ} be a discrete collection of closed subsets of X such that if α, β ∈ Γ with α 6= β then Fα 6= Fβ. For each x ∈ X, let Ux be and open neighborhood of x meeting at most one member of F. Let V be and open partial refinement of {Ux : x ∈ X} covering Y locally finite on Y. For each α ∈ Γ let Vα = ∪{V ∈ V : Y ∩ V ∩ Fα 6= φ}. Then {Vα : α ∈ Γ} is a collection of open subsets of X locally finite on Y such that for all α ∈ Γ, Y ∩ Fα ⊆ Vα ⊆ X\ ∪ (F\{Fα}). Since Y is closed and paracompact in X it is normal in X. For all α ∈ Γ let Gα and Wα be disjoint open subsets of X such that Y ∩ Fα ⊆ Gα and Y ∩ (∪(F\{Fα})) ⊆ Wα. For all α ∈ Γ let Hα = Gα ∩ Vα and Uα = Hα\∪{Hβ : β ∈ Γ\{α}}. The collection U = {Uα : α ∈ Γ} is a pairwise disjoint collection of open subsets of X. Since for all α ∈ Γ, Uα ⊆ Vα the collection U is locally finite on Y and Uα ⊆ X\ ∪ (F\{Fα}). Thus we need only show that Fα ∩ Y ⊆ Uα. Note that for all α ∈ Γ, Fα ∩ Y ⊆ Hα and since Hα ⊆ Vα the collection {Hα : α ∈ Γ} is also locally finite on Y. Thus by Lemma 2.1 the collection {Hα : α ∈ Γ} is weakly closure preserving with respect to Y and so for all α ∈ Γ Y ∩ (∪{Hβ : β ∈ Γ\{α}}) = Y ∩ (∪{Hβ : β ∈ Γ\{α}}). Suppose α ∈ Γ, x ∈ Y ∩ Fα and λ ∈ Γ\{α}. Since λ 6= α and x ∈ Y ∩ Fα, x ∈ Y ∩ (∪(F\{Fλ})) ⊆ Wλ. Since Hλ ⊆ Gλ and Gλ ∩ Wλ = φ, x /∈ Hλ. Hence (Y ∩ Fα) ∩ (∪{Hβ : β ∈ Γ\{α}}) = φ and so Y ∩ Fα ⊆ Uα. We now proceed much as in Theorem 5.1.17 of [5]. Let F = Y ∩ (∪F) and K = Y \ ∪ U. Since F and K are disjoint subsets of Y closed in X and Y is normal in X there exist disjoint open sets W and W ′such that F ⊆ W, K ⊆ W ′. Clearly for all α ∈ Γ, Y ∩ Fα ⊆ W ∩ Uα and the collection {W ∩ Uα : α ∈ Γ} is pairwise disjoint. Suppose y ∈ Y. If α ∈ Γ and y ∈ Uα then Uα is an open neighborhood of y meeting at most one member of {W ∩ Uα : α ∈ Γ}, (that member being W ∩ Uα) If y /∈ Uα for all α ∈ Γ then y ∈ K and so W ′ is an open neighborhood of y missing all members of {W ∩ Uα : α ∈ Γ}. Thus the collection {W ∩Uα : α ∈ Γ} is a pairwise disjoint collection of open sets discrete on Y such that for all α ∈ Γ, Y ∩ Fα ⊆ W ∩ Uα ⊆ X\ ∪ (F\{Fα}). � The following is a natural relative version of Bing’s Theorem. In light of Example 5.4, we need to assume that the subspace Y is relatively regular in X. Theorem 3.3. Suppose Y is strongly regular in the space X. If Y is paracom- pact in X then Y is collectionwise normal in X. Proof. Let F = {F α : α ∈ Γ} be a discrete collection of closed subsets of X such that if α, β ∈ Γ with α 6= β then Fα 6= Fβ. Using the fact that Y is strongly regular in X, for each x ∈ ∪{F α : α ∈ Γ} let Ux be an open neighborhood of x such that |{α ∈ Γ : Ux ∩ Fα 6= φ}| = 1 and |{α ∈ Γ : Ux ∩ Fα ∩ Y 6= φ}| = 1. For each x ∈ X \ ∪{F α : α ∈ Γ}, let Ux be an open neighborhood of x such 204 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir that Ux ∩ ∪{Fα ∩ Y : α ∈ Γ} = φ. Let U = {Ux : x ∈ X}. Since Y is paracompact in X, there is an open partial refinement V of U such that V covers Y and V is locally finite on Y . Note that since V is a partial refinement of U, |{α ∈ Λ : V ∩ Fα ∩ Y 6= φ}| ≤ 1 for all V ∈ V. For each y ∈ Y, let Vy ∈ V such that y ∈ Vy. For each α ∈ Γ, let Yα = Y ∩ Fα. For each y ∈ ∪{Y α : α ∈ Γ}, let Wy be an open neighborhood of y such that Wy ⊆ Vy and |{V ∈ V : Wy ∩ V 6= φ}| < ℵ0. Also, for each y ∈ ∪{Yα : α ∈ Γ}, let Oy = Wy \ ∪{V : V ∈ V, Wy ∩ V 6= ∅, and y /∈ V }. For each α ∈ Γ, let Oα = ∪{Oy : y ∈ Yα}. Clearly, Fα ∩ Y = Yα ⊆ Oα ⊆ X\ ∪ (F\{Fα}) for all α ∈ Γ. It remains to show that {Oα : α ∈ Γ} is discrete with respect to Y . To see this, let z ∈ Y, and β, γ ∈ Γ with β 6= γ. It suffices to show that either Vz ∩ Oβ = φ or Vz ∩ Oγ = φ. By the choice of Vz, either Vz ∩ Yβ = φ or Vz ∩ Yγ = φ. Without loss of generality, suppose that Vz ∩ Yγ = φ. To see that Vz ∩ Oγ = φ, let u ∈ Yγ. Either Wu ∩ Vz = φ or Ou ⊆ Wu \ Vz. In either case, Ou ∩ Vz = φ. Since u was chosen arbitrarily, Vz ∩ Oy = φ for all y ∈ Yγ. Therefore, Vz ∩ Oγ = φ, as desired. � It is not clear as to how one might modify the definition of collectionwise normality in a space X to obtain a stronger version that would be implied by being 1− paracompact in X but not by being paracompact in X. A space X is said to be discretely expandable if every discrete collection of subsets of X is expandable to a locally finite open collection, [9]. A normal space is collectionwise normal if and only if it is discretely expandable, [9]. For a space X and Y ⊆ X,we say that Y is (1−) discretely expandable in X provided every discrete collection of closed subsets of X, F there is a collection of open subsets of X, {U(F) : F ∈ F} locally finite on Y such that for all F ∈ F, Y ∩ F ⊆ U(F) ⊆ X\ ∪ (F\{F}), (F ⊆ U(F) ⊆ X\ ∪ (F\{F})). Clearly, if Y is (1−) paracompact in X then Y is (1−) discretely expandable in X. Theorem 3.4. Suppose Y is s− normal in the space X. If Y is 1− discretely expandable in X then Y is collectionwise normal in X. Proof. Let F ={Fα : α ∈ Γ} be a discrete collection of closed subsets of X such that if α, β ∈ Γ with α 6= β then Fα 6= Fβ. For each x ∈ X let U(x) be an open neighborhood of x meeting at most one member of F. Let V ={Vα : α ∈ Γ} be a collection of open subsets of X locally finite on Y such that for all α ∈ Γ, Fα ⊆ Vα ⊆ X\ ∪ (F\{Fα}). Since Y is s− normal in X, for all α ∈ Γ there exist open sets Wα and Mα such that Y ∩ Fα ⊆ Wα ⊆ Wα ⊆ Vα and Y ∩ Wα ⊆ Mα ⊆ Mα ⊆ Vα. For all α ∈ Γ let Gα = Wα\∪{Mβ ∪ Wβ : β ∈ Γ\{α}}. Note that for all α, β ∈ Γ, if α 6= β then Gα ∩ Gβ = φ. Suppose α ∈ Γ and x ∈ Fα ∩ Y. Since the collection {Mγ ∪ Wγ : γ ∈ Γ} is locally finite on Y, if x ∈ ∪{Mβ ∪ Wβ : β ∈ Γ\{α}} then x ∈ Mβ ∪ Wβ ∩ Y = (Mβ ∪Wβ)∩Y = Mβ ∩Y for some β ∈ Γ\{α}. However Mβ ⊆ Vβ and Vβ ∩Fα = φ for all β ∈ Γ\{α} a contradiction. Hence x /∈ ∪{Mβ ∪ Wβ : β ∈ Γ\{α}} and so Fα ∩ Y ⊆ Gα ⊆ X\ ∪ (F\{Fα}) for all α ∈ Γ. Relative Collectionwise Normality 205 Suppose that x ∈ Y. Since the collection {Wα : α ∈ Γ} is locally finite on Y, if x ∈ ∪{Wα : α ∈ Γ} then there is an α ∗ ∈ Γ such that x ∈ Wα∗ . Thus Mα∗ is an open neighborhood of x meeting at most one member of {Gα : α ∈ Γ}, i.e. Gα∗. Hence the collection {Gα : α ∈ Γ} is discrete with respect to Y. � Theorem 3.5. Suppose Y is closed and s− normal in the space X. If Y is discretely expandable in X then Y is strongly collectionwise normal in X. Proof. Proceed as in Theorem 3.4 replacing the closed discrete collection {Fα : α ∈ Γ} with the closed discrete collection {Y ∩ Fα : α ∈ Γ}. � Theorem 3.6. Suppose Y is strongly normal in the space X. If Y is discretely expandable in X then Y is collectionwise normal in X. Proof. Let F ={Fα : α ∈ Γ} be a discrete collection of closed subsets of X such that if α, β ∈ Γ with α 6= β then Fα 6= Fβ. Let V ={Vα : α ∈ Γ} be a collection of open subsets of X locally finite on Y such that for all α ∈ Γ, Fα ∩ Y ⊆ Vα ⊆ X\ ∪ (F\{Fα}). For all α ∈ Γ, since Y is strongly normal in X and Fα ∩ Y ⊆ Vα, there exist open sets Wα and Mα such that Fα ∩ Y ⊆ Wα ⊆ Vα, Wα ∩ Y ⊆ Mα ⊆ Vα and Mα ∩ Y ⊆ Vα. For all α ∈ Γ let Gα = Wα\∪{Mβ ∪ Wβ : β ∈ Γ\{α}}. Then as in Theorem 3.4 for all α ∈ Γ, Fα ∩Y ⊆ Gα ⊆ X\∪(F\{Fα}) and the collection {Gα : α ∈ Γ} is discrete with respect to Y. � Question 1 Suppose Y is s- normal in the space X and discretely expandable in X. Is Y collectionwise normal in X? For a normal space X, a subspace Z can be collectionwise normal in X without being 1− discretely expandable in X, Example 5.2. A subspace Y of a normal space X can be 1− paracompact in X but not strongly collectionwise normal in X. In fact a subspace of a compact space X need not be strongly collectionwise normal in X, Example 5.5. In [4] a relative property of para- compactness type which does imply strongly collectionwise normality in X is introduced. Suppose X is a set, U, V collections of subsets of X and y ∈ X. The collection V is said to star refine U at y provided there is a U ∈ U such that st(y, V) ⊆ U. For a space X, a subspace Y is said to be strongly star - normal in X provided for every collection U of open subsets of X covering Y there is a collection V of open subsets of X covering Y which star refines U at every point of ∪V. Theorem 3.7. If Y is strongly star normal in the space X then Y is strongly collectionwise normal in X. Proof. (Proceed as in Theorem 5.1.18 of [5]) Let F ={Fα : α ∈ Γ} be a col- lection of closed subsets of X which is discrete with respect to Y such that if α, β ∈ Γ with α 6= β then Fα 6= Fβ. For each y ∈ Y let Uy be and open neighborhood of y meeting at most one member of F. Let W be a collection of open subsets of X covering Y which star refines U = {Ux : x ∈ Y } at every point of ∪W and V be a collection of open subsets of X which covers 206 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir Y and star refines W at every point of ∪V. Then using the same argument as in Lemma 5.1.15 of [5], we see that V is a collection of open subsets of X covering Y such that for every V ∈ V there is a U ∈ U with st(V, V) ⊆ U. For each α ∈ Γ let Vα = ∪{V ∈ V : V ∩ Fα 6= φ} and note that for all α ∈ Γ Fα ∩ Y ⊆ Vα ⊆ X\ ∪ (F\{Fα}) and the collection {Vα : α ∈ Γ} is discrete with respect to Y. � 4. Relative versions of the Michael-Nagami Theorem By replacing “locally finite” with “point finite” in the definitions of (1−) paracompactness we obtain relative metacompact analogs [7]. The subspace Y of X is strongly metacompact in X provided every open cover of X has an open refinement point finite on Y . The subspace Y of a space X is metacompact in X provided every open cover of X has an open partial refinement point finite on Y . Clearly for a space X strongly metacompactness in X is a natural relatively metacompact analog of 1− paracompactness in X and metacompactness in X is the corresponding relative metacompact analog of paracompactness in X. Before presenting several relative versions of the Michael - Nagami Theorem here are several examples clarifying the limitations of what we can expect. A closed discrete subspace of a normal space X is always strongly metacompact in X and collectionwise normal but need not be paracompact in X, Example 5.2. In Example 5.6 we give a regular space X having an open subspace Y which is strongly collectionwise normal in X and strongly metacompact in X but not 1− paracompact in X. In Example 5.7 we give a non regular space X having a closed subspace Y which is super regular in X, strongly metacompact in X and 1− discretely expandable in X but not 1− paracompact in X. Question 2 Suppose Y is strongly metacompact in X and 1− discretely ex- pandable in the space X. Is Y paracompact in X? The proof of Theorem 5.3.3 (Michael-Nagami Theorem) of [5] can be readily modified to prove the following relative version. Theorem 4.1. Suppose that X is a regular space and Y ⊆ X. If Y is strongly metacompact in X and 1− discretely expandable in X then every open cover of X has an open partial refinement covering Y which is the countable union of collections locally finite on Y. Question 3 Suppose that X is a regular space, Y ⊆ X and every open cover of X has an open partial refinement covering Y which is the countable union of collections locally finite on Y. Is Y paracompact in X? For a closed subspace Y of a space X, Y is paracompact in X if and only if every open cover of X has an open partial refinement covering Y which is the countable union of collections locally finite on Y, [8]. Also for a closed subset Y, Y is strongly metacompact in X if and only if Y is a metacompact subspace of X, [7]. Relative Collectionwise Normality 207 Corollary 4.2. Suppose Y is closed in the regular space X. If Y is 1- discretely expandable and metacompact then Y is paracompact in X. (Is Y 1- paracompact in X?) Question 4 Suppose Y is (strongly) metacompact in X and collectionwise normal in X. Is Y paracompact in X? In Question 3 if locally finite on Y is replaced with discrete with respect to Y the answer is yes. Lemma 4.3. Suppose that Y is strongly regular and strongly collectionwise normal in the space X. If every open cover of X has an open partial refinement covering Y which is the countable union of collections discrete with respect to Y then Y is paracompact in X. Proof. Let U be an open cover of X. For all x ∈ X let Wx be an open neigh- borhood of x such that Wx ⊆ U and Y ∩ Wx ⊆ U for some U ∈ U. Let W = {Wx : x ∈ X} and V = ∪{Vn : n < ω} be an open partial refine- ment of W covering Y such that for all n < ω, the collection Vn is discrete with respect to Y. For all n < ω, since Vn is discrete with respect to Y, the collection {V : V ∈ Vn} is discrete with respect to Y. For each n < ω let Gn = {G(V, n) : V ∈ Vn} be a collection of open subsets of X discrete with re- spect to Y such that for all V ∈ Vn, V ∩Y ⊆ G(V, n) and G(V, n) ⊆ U for some U ∈ U. For each n < ω let Fn = ∪Vn. For each V ∈ Vo let H(V, 0) = G(V, 0). For each 0 < n < ω and V ∈ Vn let H(V, n) = G(V, n)\ ∪ {Fk : k < n}. For each n < ω let Hn = {H(V, n) : V ∈ Vn} and let H = ∪{Hn : n < ω}. We now show that H covers Y and is locally finite with respect to Y. Let y ∈ Y. Let n = min{k < ω : y ∈ Fk}. Since y ∈ Y ∩ Fn and Vn is discrete with respect to Y, there is a V ∈ Vn with y ∈ V ∩ Y ⊆ G(V, n) and so y ∈ H(V, n). Let m = min{k < ω : y ∈ ∪Vk} and V ′ ∈ Vm such that y ∈ V ′. Since V ′ ⊆ Fm, V ′ is an open neighborhood of y missing all members of Hk for all m < k < ω. For all k ≤ m, since the collection Gk and hence Hk is discrete with respect to Y, let Ok be an open neighborhood of y meeting at most one member of Hk. Then V ′ ∩ Oo ∩ ... ∩ Om is an open neighborhood of y meeting only finitely many members of H. � Again the proof of Theorem 5.3.3 (Michael-Nagami Theorem) of [5] can be readily modified to prove the following relative version. We include a proof here to demonstrate the modifications needed for this theorem and in the proof of Theorem 4.1. Theorem 4.4. Suppose that Y is strongly regular in the space X. If Y is meta- compact in X and strongly collectionwise normal in X then Y is paracompact in X. Proof. Let O be an open cover of X and let U = {Uα : α ∈ Γ} be an open partial refinement of O covering Y point finite on Y such that if α, β ∈ Γ and α 6= β then Uα 6= Uβ. Let Vo = {φ}. Suppose k < ω and for all i ≤ k the collection Vi has been defined and Wi = ⋃ Vi such that 208 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir 1. Vi is an open partial refinement of U discrete with respect to Y 2. if x ∈ Y such that |{α ∈ Γ : x ∈ Uα}| ≤ i then x ∈ ∪{Wj : j = 0..i}. Let Tk+1 = {T ⊆ Γ : |T | = k + 1} and for all T ∈ Tk+1 let FT = (X\ ∪ {Wj : j = 0..i}) ∩ (X\ ∪ {Uα : α ∈ Γ\T }). Suppose T ∈ Tk+1. If x ∈ Y ∩FT then {α ∈ Γ : x ∈ Uα} ⊆ T and x /∈ ∪{Wj : j = 0..k}. Hence {α ∈ Γ : x ∈ Uα} = T and so Y ∩ FT ⊆ ∩{Uα : α ∈ T }. Suppose that x ∈ Y. If |{α ∈ Γ : x ∈ Uα}| ≤ k then ∪{Wj : j = 0..k} is an open neighborhood of x missing all members of {FT : T ∈ Tk+1}. Suppose |{α ∈ Γ : x ∈ Uα}| ≥ k + 2. Let α1, α2, ..., αk+2 be distinct members of {α ∈ Γ : x ∈ Uα}. Then ∩{Uαi : i = 1..k + 2} is an open neighborhood of x meeting no member of {FT : T ∈ Tk+1}. Suppose |{α ∈ Γ : x ∈ Uα}| = k + 1. Let T ′ = {α ∈ Γ : x ∈ Uα} and note ∩{Uα : α ∈ T ′} is a neighborhood of x meeting exactly one member of {FT : T ∈ Tk+1}. Hence we see that {FT : T ∈ Tk+1} is a collection of closed subsets of X which is discrete with respect to Y. Let {GT : T ∈ Tk+1} be a collection of open subsets of X discrete with respect to Y such that for all T ∈ Tk+1, Y ∩ FT ⊆ GT ⊆ X\(∪{FT ′ : T ′ ∈ Tk+1\{T }}). Also assume that for all T ∈ Tk+1, if Y ∩ FT = φ then GT = φ. For all T ∈ Tk+1, let VT = GT ∩ (∩{Uα : α ∈ T }) and note that Y ∩ FT ⊆ VT . Let Vk+1 = {VT : T ∈ Tk+1} and Wk+1 = ⋃ Vk+1. Then Vk+1 is an open partial refinement of U discrete with respect to Y. Suppose that x ∈ Y such that |{α ∈ Γ : x ∈ Uα}| ≤ k + 1. Then there is a T ∈ Tk+1 such that x ∈ X\ ∪ {Uα : α ∈ Γ\T }. Thus x ∈ X\ ∪ {Uα : α ∈ Γ\T } = ((X\ k ⋃ i=0 Wi) ∪ ( k ⋃ i=0 Wi)) ∩ (X\ ∪ {Uα : α ∈ Γ\T }) = [(X\ k ⋃ i=0 Wi) ∩ (X\ ∪ {Uα : α ∈ Γ\T })] ∪ [( k ⋃ i=0 Wi) ∩ (X\ ∪ {Uα : α ∈ Γ\T })] ⊆ FT ∪ k ⋃ i=0 Wi. Hence for all x ∈ Y such that |{α ∈ Γ : x ∈ Uα}| ≤ k + 1, x ∈ k+1 ⋃ i=0 Wi. Thus, since U is point finite on Y, V = ⋃ n<ω Vn is an open partial refinement of U covering Y such that for all n < ω the collection Vn is discrete with respect to Y. By Lemma 4.3, Y is paracompact in X. � Corollary 4.5. Suppose Y is closed and s− regular in the space X. Then Y is paracompact in X if and only if Y is collectionwise normal in X and metacompact. 5. Examples Example 5.1. A T2 space X having a subspace Y which is 1− paracompact in X but not strongly regular in X. Relative Collectionwise Normality 209 Let X = ω ∪ (ω × ω) ∪ {∗}. Define a topology on X as follows: 1. points of ω × ω are isolated, 2. for each n < ω, {{n} ∪ ({n} × (k, ω)) : k < ω} is a local base at n, 3. the collection {{∗} ∪ ((k, ω) × ω) : k < ω} is a local base at ∗. Then X is T2 and the subspace Y = ω is 1− paracompact in X but the closed set Y cannot be separated from the point ∗ by open subsets of X. Thus Y is not strongly− regular in X. Example 5.2. Bing’s Example G. Let X be Bing’s Example G, Y the nonisolated points of X and Z the isolated points of X. The subset Y is a closed discrete subspace of X and therefore is strongly metacompact in X and collectionwise normal. However Y is not collectionwise normal in X. The subspace Z is an open discrete subspace of X and therefore strongly collectionwise normal in X and paracompact in X but not 1− discretely expandable in X. Example 5.3. A regular space X having an open normal subspace which is collectionwise normal in X but which is not 1− discretely expandable in X and a closed subspace Z which is paracompact in X but not s− normal in X. The space X is a standard modification of the Tychonoff plank. Let X = [0, ω1] × [0, ω]\{(ω1, ω)}. Define a topology on X as follows: 1. Points of ω1 × ω are isolated. 2. For all n < ω let {B(α, n) : α < ω1} be a neighborhood base for the point (ω1, n) where B(α, n) = (α, ω1] × {n} for all α < ω1. 3. For all α < ω1 let {G(α, n) : n < ω} be a neighborhood base for the point (α, ω) where G(α, n) = {ω1} × (n, ω] for all n < ω. Clearly X is a regular space. Let Y = X\({ω1}×ω). Since Y is an open normal subspace of X it is strongly normal in X. The closed sets ω1 × {ω} and {ω1} × ω cannot be separated by open subsets of X. Thus not only is X not normal but Y is not s− normal in X. The subset Y is collectionwise normal in X since it is an open subset of X and the direct sum of compact subspaces (Y = ⊕{{α} × [0, ω] : α < ω1}). However Y is not 1− discretely expandable in X. To see this let C = {ω1} × ω and F = {{r} : r ∈ C} and note that F is a discrete collection of closed subsets of X. Suppose that for all r ∈ C, U(r) is an open neighborhood of r. For all n < ω let βn < ω1 such that B(βn, n) ⊆ U(ω1, n). Let β ∗ = sup{βn : n < ω} and note that β∗ < ω1. Choose β ∗ < γ < ω1 and let k < ω. Then (γ, m) ∈ G(γ, k) ∩ B(βm, m) ⊆ G(γ, k) ∩ U(ω1, m) for all k < m < ω. Hence every neighborhood of the point (γ, ω) meets infinitely many members of {U(r) : r ∈ C}. Thus the collection {U(r) : r ∈ C} is not locally finite on Y. Let Z = {ω1} × ω. The closed discrete subspace Z is easily seen to be paracompact in X but like Y it is not s− normal in X. Example 5.4. A T2 Lindelöf space X having a subspace which is 1− para- compact in X but not collectionwise normal in X. 210 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir Let Y and Z be disjoint subsets of R\Q such that for every nonempty open subset U of R |U ∩ Y | = ω1 = |U ∩ Z|. Well order Q, Y , and Z, say Q ={qn : n < ω} , Y = {yα : α < ω1} and Z = {zα : α < ω1}. For any set A ⊆ R let qA = {n < ω : qn ∈ A}, yA = {α < ω1 : yα ∈ A} and zA = {α < ω1 : zα ∈ A}. Let X = (R × {0, 1}) ∪ (Y ∪ Z ∪ Q) ∪ (ω1 × ω × {0, 1}) and define a topology on X as follows: 1. All points of ω1 × ω × {0, 1} are isolated. 2. For all α < ω1 a basic open neighborhood of yα [zα] is of the form {yα} ∪ ({α} ×q U × {0}) [{zα} ∪ ({α} ×q U × {1})] where U is an open neighborhood of yα [zα] in R. 3. For all n < ω a basic open neighborhood of qn is of the form {qn} ∪ ((α, ω1) × {n} × {0, 1}) where α < ω1. 4. For all x ∈ R a basic open neighborhood of (x, 0) [(x, 1)] is of the form ([x, a) × {0}) ∪ ((x, a) ∩ (Y ∪ Q)) ∪ (y(x, a) ×q (x, a) × {0}) ∪((α, ω1) ×q (x, a) × {0, 1}) where a ∈ R , x < a and α < ω1. [ ((b, x] × {1}) ∪ ((b, x) ∩ (Z ∪ Q)) ∪ (z(b, x) ×q (b, x) × {1}) ∪((β, ω1) ×q (b, x) × {0, 1}) where b ∈ R , b < x and β < ω1. ] The space X is T2 Lindelöf but not regular. The subspace Y ∪ Z ∪ Q is 1− paracompact in X but not collectionwise normal in X. To see that Y ∪Z ∪Q is not collectionwise normal in X let F = (R×{0})∪Y and K = (R×{1})∪Z. Note F and K are disjoint closed subsets of X. Suppose that U and V are disjoint open subsets of X such that F ∩ (Y ∪ Z ∪ Q) = Y ⊆ U and K ∩ (Y ∪ Z ∪ Q) = Z ⊆ V. Then U ∩ V ∩ Q 6= φ. Example 5.5. A compact space X having a subspace Y which is not strongly collectionwise normal in X. Let X = (ω1 + 1) × (ω + 1) with the product topology and Y = X\{(ω1, ω)} (Tychonoff plank). Then since X is compact Y (and every other subspace of X) is 1− paracompact in X. The collection of closed subsets of X F ={(ω1 + 1) × {ω}} ∪ {{(ω, n)} : n < ω} is discrete with respect to Y . Using the same argument that the Tychonoff plank is not normal using the closed (in Y ) sets ω1 × {ω} and {ω1} × ω, one can use F to show that Y is not strongly collectionwise normal in X. Example 5.6. A regular space having a subspace which is strongly meta- compact in X and strongly collectionwise normal in X but not 1− discretely expandable in X. Let X = R × R, Y = R × {0} and Z = X\Y. Points of Z have their usual open neighborhoods. For each x ∈ R a basic neighborhood of (x, 0) will be of the form {x} × (−ǫ, ǫ) where ǫ > 0. Clearly X is regular and Z is strongly metacompact in X and strongly star normal in X. However the points of the closed discrete subset Y cannot be separated by open subsets of X which are discrete with respect to Z. Relative Collectionwise Normality 211 Example 5.7. A nonregular space having a subspace which is super regular in X, strongly metacompact in X and 1− discretely expandable in X but not 1− paracompact in X. Let A = ω1 with the order topology. Let B = [0, 1] with points of (0, 1] having usual open neighborhoods in [0, 1] with the order topology and open neighborhoods of 0 are of the form U\{ 1 n : n = 1, 2, ...} where U is a usual open neighborhood of 0 in [0, 1] with the order topology. Note that B is T2 but not regular. Also { 1 n : n = 1, 2, ...} is closed, 1− discretely expandable in B and super regular in B. The construction of the space X is based on examples in [2] and [7]. Let X = A × B with the topology defined as follows: 1. for a ∈ A and y ∈ (0, 1] basic open neighborhoods are of the form {a} × V where V is an open neighborhood of y in B, 2. for a ∈ A basic open neighborhoods of (a, 0) are of the form ∪{{x} × Vx : x ∈ U} where U is an open neighborhood of a in A and for all x ∈ U, Vx is an open neighborhood of 0 in B. Let Y = A × { 1 n : n = 1, 2, ...} and note that Y is a closed discrete subset of X and therefore strongly metacompact in X. Also note that Y is super regular in X but not strongly regular in X. It is not difficult to show that Y is 1− discretely expandable in X. To see that Y ia not 1− paracompact in X, let U = {[0, α] × B : α < ω1}. Using the Pressing Down Lemma it is easily seen that U does not have an open refinement that is locally finite on Y. References [1] A. Arhangel’skii, “From classic topological invariants to relative topological properties”, Scientiae Math. Japonicae 55 No 1 (2002), 153-201. [2] A. Arhangel’skii and H Genedi, “Beginnings of the theory of relative topological proper- ties”, Gen. Top. Spaces and Mappings (MGU, Moscow, 1989), 3-48 (in Russian). [3] A. Arhangel’skii and H Genedi, “Position of subspaces in spaces: relative versions of compactness, Lindelöf properties, and separation axioms”, Vestnik Moskovskogo Uni- versiteta, Mathematika 44 No. 6 (1989), 67-69. [4] A. Arhangel’skii and I. Gordienko, “Relative symmetrizability and metrizability”, Com- ment. Math. Univ. Carol. 37 No 4 (1996), 757-774. [5] R. Engelking, “General Topology” (PWN, Warsaw, 1977). [6] I. Gordienko, “On Relative Properties of Paracompactness and Normality Type”, Moscow Univ. Nath. Bul. 46 No. 1 (1991), 31-32. [7] E. Grabner, G. Grabner and K. Miyazaki, “ Properties of relative metacompactness and paracompactness type”, Topology Proc. 25 (2000), 145-178. [8] K. Miyazaki, “On relative paracompactness and characterizations of spaces by relative topological properties”, Math. Japonica 50 (1999), 17-23. [9] J.C. Smith and L.L. Krajewski, “Expandability and collectionwise normality”, Trans. Amer. Math. Soc. 160 (1971), 437-451. [10] Y. Yasui, “Results on relatively countably paracompact spaces”, Q and A in Gen. Top. 17 (1999), 165-174. 212 E. Grabner, G. Grabner, K. Miyazaki and J. Tartir Received April 2003 Accepted July 2003 Elise Grabner (elise.grabner@sru.edu) Dept. Math., Slippery Rock University, Slippery Rock PA, 16057 USA Gary Grabner (gary.garbner@sru.edu) Dept. Math., Slippery Rock University, Slippery Rock PA, 16057 USA Kazumi Miyazaki (BZQ22206@nifty.ne.jp) Dept. Math., Osaka Elector-Communication University, Osaka 572-8530, JAPAN Jamal Tartir (tartir@math.ysu.edu) Dept. Math. and Stat., Youngstown State University, Youngstown OH, 44555 USA