Paracompact Subspaces in the Box Product Topology 0

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 1, January 1996

PARACOMPACT SUBSPACES IN THE BOX PRODUCT TOPOLOGY

PETER NYIKOS AND LESZEK PIA¸ TKIEWICZ

(Communicated by Franklin D. Tall)

Abstract. In 1975 E. K. van Douwen showed that if (Xn)n ω is a family of ∈ Hausdorﬀ spaces such that all ﬁnite subproducts n

Theorem. Let κ be an infinite cardinal number, and let (Xα)α κ be a family ∈ of compact Hausdorff spaces. Let x = α κXα be a fixed point. Given a ∈ ∈ family of open subsets of which covers σ(x), there exists an open locally finite inR refinement of which covers σ(x). S R We also prove a slightly weaker version of this theorem for Hausdorff spaces with “all finite subproducts are paracompact” property. As a corollary we get an affirmative answer to van Douwen’s question.

0. Introduction

A box product is a topological space which takes a Cartesian product of spaces for the point-set, and takes an arbitrary Cartesian product of open subsets for a base element. The box product topology is nontrivial in the case of infinitely many factor spaces, strictly stronger than the usual Tychonoff topology of pointwise convergence, because each factor of a basic open set is permitted to be a proper subset of a factor space. In 1991 Brian Lawrence showed in [L] that (in ZFC) the box product ω 1 (ω +1)ofω1 many copies of ω + 1 is neither normal nor collectionwise Haus- dorff (and hence not paracompact), solving an old problem due to Arthur H. Stone (1964,[K]) and Mary Ellen Rudin (1975,[R]). In the same paper he proved that if 2ω =2ω1, then the Σ-product, that is,

ω1 x (ω +1): α ω1 :x(α)=ω is countable , { ∈ { ∈ 6 } } is nonnormal.

Received by the editors June 9, 1993. 1991 Mathematics Subject Classiﬁcation. Primary 54D18; Secondary 54B10. Key words and phrases. Paracompact space, box product. The ﬁrst author’s research was supported in part by NSF Grant DMS-8901931.

c 1996 American Mathematical Society

303

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In this paper we consider σ-products, where for given x in a box product α κXα the σ-product is the set ∈

σ(x)= y α Xα : α κ:y(α)=x(α) is ﬁnite . { ∈ ∈κ { ∈ 6 } }

We prove (in ZFC) that if (Xα)α is a family of topological spaces, such that ∈κ α θ Xα is paracompact Hausdorff for every finite θ κ,thenσ(x) is paracompact ∈ ⊆ for each x α κXα. This answers affirmatively a question asked by Eric van DouwenQ in∈ (1975,[vD])∈ (question 145 on van Mill’s list of van Douwen’s problems in (1993,[vM])). If each Xα is a compact Hausdorff space, we get the stronger result stated in the abstract. This result is a generalization of a theorem of Scott W. Williams (1990,[W]), who has a similar result for compact metrizable spaces.

1. Notation and terminology Let X be a topological space. (X) denotes the power set of X.Foreach A X, A denotes the closure of APin X,andint(A) denotes the interior of A in X.⊆ The family of subsets of X is a cover (or covering)ofXif = X. is an open coveringWof X if each set in is an open subset of X. isW an irreducibleW open covering of X if is an openW covering of X which is irreducible,WS that is, no proper subfamily of Wcovers X. W For given infinite cardinal κ and a family of topological spaces Xα : α κ , { ∈ } = α Xα denotes the box product of Xα’s. If λ κ,thenProj λ is the ∈κ ⊆ projection from α Xα onto α λXα.Ifx ,thenabox neighborhood of x ∈κ ∈ ∈ is any set of the form α Aα where x(α) int(Aα)foreachα κ.Thebox ∈κ ∈ ∈ neighborhood Aα is open if each Aα is an open subset of Xα. α κ The σ-product ∈σ(x)Q is the following set: Q σ(x)= y : α κ:x(α)=y(α) is finite . { ∈ { ∈ 6 } } [κ]<ω is the family of all finite subsets of κ.Let and be two families of subsets T S of .Wesaythat refines if for each T there is S such that T S; we say that strictlyT refinesS if for each T ∈T there is S ∈S such that T ⊆ S. T S ∈T ∈S ⊆ Note that we do not require that = . For every finite λ κ we will use the following notation: T S ⊆ S S λ = α λXα, ∈ λ for each y ,y =Proj λ(y), ∈ for A ,Aλ = yλ :y A , ⊆ ∈ λ λ for ( ), =A :A . A⊆P A ∈A 2. The paracompact case Here is our main result.

Theorem 2.1. Let κ be an infinite cardinal and let Xα : α κ be a family of { ∈ } spaces such that λ is paracompact Hausdorff for each λ [κ]<ω.Letx = ∈ ∈ α Xαbe a fixed point and let be a family of open sets in which covers σ(x). ∈κ R

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There exist an open in neighborhood U of σ(x) and a locally finite in U family of open sets, such that strictly refines and = U. B B R B Note that the conclusion is strictly strongerS than the claim that σ(x) is paracompact. For example, a discrete space is paracompact, yet even a closed discrete subspace of a regular space need not have the property in the conclusion. Indeed, if one covers the x-axis in the tangent disk space with basic tangent disk neighborhoods at each point, no such U or exist. Before proving the theorem weB first prove a rather technical lemma. In it, we produce open covers of σ(x) by induction: first of x itself, then all points that deviate from it in exactly one coordinate, then two coordinates, and so forth. At the nth stage of the induction, we take each n-element subset λ of κ, and handle all the points which deviate from x at exactly the coordinates indexed by λ.Thisset λ of points is homeomorphic to so a lot of our attention is focused on producing λ λ λ the projections ( λ) ,( λ) and ( λ) of the covers λ, λ and λ for this set, W Vλ λU W V U which we identify with xκ− . × Lemma 2.2. Under the assumptions of Theorem 2.1, there are collections

# <ω <ω = : λ [κ] , = λ : λ [κ] W Wλ ∈ U U ∈ [ n o [ and <ω = λ : λ [κ] , V V ∈ of open subsets of subject to the[ following conditions for all finite λ κ: # λ ⊆ (A) (a0) If λ = : τ λ ,then( λ) is a locally finite open cover of W Wτ ⊆ W λ, such that for each W : λ (a1) There exists RS ∈W, such that W R, λ ∈Rλ λ ⊆ λ (a2) W = W W κ− where W κ− is an open box neighborhood of xκ− . × λ λ For each W λ we choose an open box neighborhood U(W )κ− of xκ− ,so that: ∈W κ λ λ (a3) U(W ) − W κ− . λ ⊆ λ (B) (b0) ( λ) is a locally finite open cover of and the following conditions U are satisfied for each U λ: ∈U (b1) There exists W λ, such that U W , λ λ∈W ⊆ (b2) U = U U κ− , × λ λ (b3) (U)= W λ :U W =∅ is a finite set, STWλ ∈W λ ∩ 6 (b4) U κ− = U(W )κ− : W (U) . λ ∈STW λ (C) (c0) ( λ) is a locally finite open cover of and the following conditions V T are satisfied for each V λ: ∈V (c1) There exists U λ, such that V U, λ ∈Uλ ⊆ (c2) V = V V κ− , × λ λ (c3) (V )= U λ :V U =∅ is a finite set, STU λ ∈Uλ ∩ 6 (c4) V κ− = U κ− : U (V) . U Let’s note that (b3) together∈ST with (c3) give us: 2 T (c5) (V )= (U):U (V) is finite, for each V λ. ST {STW ∈STU } ∈V (D) For each W # and each θ λ,thereexistsV , such that the S0 λ ( 0 θ following conditions hold∈W: ∈V θ θ (d1) (W0) (V0) , ⊆

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λ θ λ θ (d2) (W0) − is disjoint from U(W ) − : W (V0) , ∈STW λ λ 2 (d3) (W0)κ− U(W )κ− : Wn (V0) . o ⊆ T ∈ST Proof of Lemma 2.2T. We construct families

# <ω <ω = : λ [κ] , = λ : λ [κ] W Wλ ∈ U U ∈ [ n o [ and <ω = λ : λ [κ] V V ∈ by induction on n = λ .Forn[= 0 choose any open box neighborhoods W0 | | and U(W0)ofx,sothatU(W0) W0 and W0 R,forsomeR . We put ⊆ ⊆ ∈R # = W and = = U(W ) . Clearly all conditions in (A)-(D) are ∅ 0 ∅ ∅ 0 satisﬁed.W { } U V { } # Now assume that for some natural number n, θ , θ, θ have been W U V θ n | |≤ constructed so that all the conditions in (A)-(D) hold,n and let λ o[κ]n+1.Foreach ∈ θ(λlet

θ λ θ (1) θ = V U(W ) − : W (V) :V θ , F × ∈STW ∈V n \ n o o and let

(2) F = θ : θ ( λ . F [ n[ o By (c0) for θ ( λ we get:

λ (3) each θ is a locally finite family of closed subsets of . F By (c1),(b1),(a3) for θ ( λ and the definition of in (b3) we get: STW λ (4) For each θ ( λ, θ ( θ) . F ⊆ W [ [ Now (2),(3) and (4) together with the definition of θ in (a0) give us: W λ # λ (5) F is a closed subset of contained in ( ) : θ ( λ . Wθ [ n[ o For each θ ( λ let

θ λ θ (6) θ = V − : V θ . O × ∈V λ By (c0) for θ ( λ each θ is an open covering of .Let O λ # λ (7) X = ( ) : θ ( λ . \ Wθ n[ o F and X are disjoint closed subsets (see (5) and (7)) of a paracompact space λ, hence there exists a family ( #)λ of open subsets of λ, such that: Wλ # λ (8a) For each θ ( λ, ( ) is a locally ﬁnite reﬁnement of θ, Wλ O

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(8b) ( #)λ covers X, Wλ

(8c) ( #)λ is disjoint from F, Wλ [

λ # λ λ κ λ λ λ (8d) W ( ) RW such that xκ− R − and W R . ∀ ∈ Wλ ∃ ∈R ∈ W ⊆ W

λ # λ κ λ κ λ For each W ( λ ) we choose an open box neighborhood W − of x − ,so that: ∈ W

λ κ λ κ λ λ θ θ (9) W κ− R − V − : V θ,θ(λand (W ) V , ⊆ W ∩ ∈V ⊆ \ λ λ and choose an open box neighborhood U(W )κ− of xκ− so that

κ λ λ (10) U(W ) − W κ− . ⊆ Let

# λ λ λ # λ (11) = W W κ− : W ( ) . Wλ × ∈ Wλ n o # λ λ Note that the family ( ) was already chosen (see 8(a-d)), and that W κ− Wλ has been chosen for each W λ ( #)λ (see (9)), so this is not circular. ∈ Wλ λ # λ Let us check that ( λ) = ( ) : θ λ satisfies the conditions in (A) W Wθ ⊆ and (D). n o S λ First (8a) together with (a0) for θ ( λ give us that ( λ) is locally finite, while λ W (7) and (8b) imply that it covers , hence (a0) holds. Now (8d),(9) and (11) imply (a1), (11) implies (a2) and (10) gives us (a3). Next (6) and (8a) imply (d1). By (1), (2) and (8c) we get (d2), while (9) and 2 (d1) together with conditions (b4) and (c4) for θ ( λ and the definition of in (c5) give us (d3). Thus all the conditions in (A) and (D) are satisfied. ST λ λ Finally since ( λ) is a locally finite open covering of a paracompact space , W we can choose open locally finite families λ and λ, so that all conditions in (B) U V and (C) are also satisfied. We are ready now to prove the theorem.

Proof of Theorem 2.1. Let be a family of open sets in which covers σ(x)and strictly reﬁnes . T We constructR families , and for as in Lemma 2.2. W U V T # # Lemma A. λ, θ [κ]<ω if λ * θ and θ * λ,then =∅. ∀ ∈ Wλ ∩ Wθ <ω Proof of Lemma A. Assume that λ, θ [κ] are suchS that λS* θ and θ * λ. # # ∈ Let W1 ,W2 and suppose that: ∈Wλ ∈Wθ

(A1) W1 W2 = ∅. ∩ 6

Let δ = λ θ.Sinceδ(λand δ ( θ,wecanchooseV1,V2 δ,sothat Wδ Vδ,W∩δ Vδ and the conditions (d2) and (d3) are satisﬁed∈V (see (D) in 1 ⊆ 1 2 ⊆ 2

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δ δ Lemma 2.2). Clearly V1 V2 = ∅.Since reﬁnes we get (see the deﬁnitions of and 2 in (c3) and∩ (c5)6 in LemmaV 2.2): U STW ST 2 (A2) (V1) (V2). STW ⊆ST By (d2) in Lemma 2.2 we get:

λ δ λ δ (A3) (W1) − is disjoint from U(W ) − : W (V1) . ∈STW \ n o Since (λ δ) (κ θ), (A2) above and (d3) in Lemma 2.2 imply that: − ⊆ − λ δ λ δ 2 (W2) − U(W ) − : W (V2) (A4) ⊆ ∈ST λ δ \ U(W) − :W (V1 ) . ⊆ ∈STW \ λ δ λ δ By (A3) and (A4) we get (W1) − (W2) − = ∅ which contradicts (A1). ∩ The following part of the proof mimics a well-known argument, used to show that if every open cover of a regular space has a σ-locally finite open refinement which covers the space, then every open cover of the space has a locally finite closed refinement which covers the space. See for example [E]. For each n ω put: ∈ n = λ : λ = n , W {W | | } [

(1) On = n m : m

(2) n = W On : W n . F ∩ ∈W Put = n : n ω . F {F ∈ } Note that (see (a1) in Lemma 2.2)[

(3) F = W On TF such that F W TF . ∀ ∩ ∈F ∃ ∈T ⊆ ⊆ Clearly (see (b0),(b1) in Lemma 2.2 and (1),(2) above)

(4) σ(x) . ⊆ U⊆ W⊆ F [ [ [ <ω Lemma B. Each U = θ : θ [κ] meets only ﬁnitely many sets in . ∈U {U ∈ } F <ω Proof of Lemma B. Fix U0S θ for some θ [κ] .Since θreﬁnes θ = # ∈U # ∈ U W : τ θ ,wecanchooseW0 for some τ0 θ,sothat: Wτ ⊆ ∈Wτ0 ⊆ S (B1) U0 W0. ⊆

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Let m = τ0 ,andletF meet U0, | | ∈F

(B2) F U0 = ∅. ∩ 6 # Say F = W On for some W with λ0 = n.SinceU0is an open subset of ∩ ∈Wλ0 | | , (B2) above implies that W On U0 = ∅, hence On U0 = ∅ and by (B1) we get: ∩ ∩ 6 ∩ 6

(B3) On W0 = ∅. ∩ 6

Also W U0 = ∅ and again by (B1) we have: ∩ 6

(B4) W W0 = ∅. ∩ 6

Now (1) and (B3) imply that n m, while (B4) implies that τ0 λ0 or λ0 τ0 ≤ ⊆ ⊆ (see Lemma A). Thus λ0 τ0 θ and the following inclusions hold: ⊆ ⊆

F = W On : F U0 = ∅ ∩ ∩ 6 # F = W On : n m, W with λ = n, λ θ, and U0 W = ∅ ⊆ ∩ ≤ ∈Wλ | | ⊆ ∩ 6 n θ θ o F = W On : n m, W θ and U0 W = ∅ . ⊆ ∩ ≤ ∈W ∩ 6 By (b3) in Lemma 2.2 the last set is finite, hence the first set is finite as well. The following part of the proof mimics a standard argument, used to show that if every open cover of a regular space has a locally finite closed refinement which covers the space, then the space is paracompact (see [E]). is a family of open subsets of with σ(x) , so we can apply Lemma U ⊆ U 2.2 to to get families 0, 0, 0 and then 0, just like , , and were constructedU for .ForeachW FU V put F S W U V F T ∈F

(5) F ∗ = 0 F 0 0 :F0 F =∅ , U \ { ∈F ∩ } [ [ and let ∗ = F ∗ : F . F { ∈F} Note that for each F the following inclusions hold: ∈F

(6) 0 F F ∗ 0, U ∩ ⊆ ⊆ U [ [ and since 0 reﬁnes , we get that (see (4)) 0 , hence U U U ⊆ F S S (7) ∗ = 0. F U [ [ By Lemma B we get F 0 0 : F 0 0 is a locally ﬁnite in 0 family of { ∩ U ∈F} U closed subsets of 0, hence (see (5)) U S S S (8) ∗ is an open cover of 0. F U [

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Lemma C. ∗ is locally finite in 0. F U Proof of Lemma C. Let U 0 0. WeS will show that U 0 meets only finitely many ∈U sets in ∗. First let us notice that Lemma B implies that U 0 meets only finitely F many elements of 0. Let F

F10,F20,...,Fn0 = F0 0 :F0 U0 =∅ . { } { ∈F ∩ 6 }

Note that 0 0 (see (4)), hence U ⊆ F S S (C1) U 0 F 0 : i n . ⊆ { i ≤ } [ For each i n ﬁx Ui ,sothat(see(3)) ≤ ∈U

(C2) F 0 Ui. i ⊆

For each F ∗ ∗ we have: ∈F

U 0 F ∗ =∅ ∩ 6 = i nFi0 F∗=∅(see (C1)), ⇒∃≤ ∩ 6 = i nFi0 F=∅(see (5)), ⇒∃≤ ∩ 6 = i nUi F=∅(see (C2)). ⇒∃≤ ∩ 6

Now Lemma B implies that each F :Ui F =∅ is finite, hence { ∈F ∩ 6 } F ∗ ∗ :U0 F∗ =∅ is a finite set as well. { ∈F ∩ 6 } For each F choose TF ,sothatF TF (see (3)) and put BF = F ∗ TF . ∈F ∈T ⊆ ∩ Let = BF : F . We will show that 0 and satisfy all the required conditions.B { ∈F} U B S First since 0 was constructed for in the same way as was constructed for U U U , (4) implies that σ(x) 0. It is also clear that consists of open sets (see (8)).T ⊆ U B Now since strictly refinesS we get T R

BF = F ∗ TF R such that BF TF R. ∀ ∩ ∈B∃ ∈R ⊆ ⊆

Each BF is a subset of corresponding F ∗ so Lemma C implies that the family is B locally ﬁnite in 0. U Finally ∗ and since 0 F F ∗ TF = BF for each F (see B⊆S F U ∩ ⊆ ∩ ∈F (3) and (6)), we get 0 ∗. On the other hand, (7) and (4) S S U ∩ F⊆S B⊆ F imply that ∗ = 0 and so = ∗ = 0. F SU ⊆ SU⊆ SF S B F U This concludes the proof of Theorem 2.1. S S S S S S S As a corollary to Theorem 2.1 we obtain the following aﬃrmative answer to van Douwen’s question.

Corollary 2.3. If (Xα)α is an uncountable family of spaces such that all ﬁnite ∈κ subproducts are paracompact Hausdorﬀ, then for each x α κXα the σ-product, σ(x), is paracompact. ∈ ∈

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3. The compact case

If each Xα is a compact Hausdorﬀ space, we have a strengthening of Theorem 2.1 mentioned in the abstract. We also have the following nicer version of Lemma 2.2.

Lemma 3.1. Under the assumptions of Theorem 2.1 if each Xα is a compact Hausdorﬀ space, there are families

<ω <ω = Uλ : λ [κ] and = Vλ : λ [κ] U ∈ V ∈ of open box neighborhoods of x,andacollection

# <ω = :λ [κ] W Wλ ∈ n o of ﬁnite families of open boxes in , subject to the following conditions for all λ [κ]<ω: ∈ # λ λ (i) λ = : τ λ is an irreducible open covering of xκ− . W Wτ ⊆ × (ii) W # R such that W R. ∀ ∈WS λ ∃ ∈R ⊆ (iii) α/λ Uλ α Vλ Vλ Uλ. ∀ ∈ ∪{ } ⊆ ⊆ ⊆ (iv) W W # ∀ ∈ λ (a) α/λW(α)=Uλ(α), ∀ ∈ (b) τ ( λ β λ τ such that W (β) Vτ (β)=∅. ∀ ∃ ∈ − ∩ ProofofLemma3.1. We construct families , , and by induction on n = λ . U V W | | For n = 0 pick any open box neighborhoods of x, U∅ and V∅,sothatx V∅ # ∈ ⊆ V U U Rfor some R , and put = U . Clearly all four ∅ ∅ ∅ ∅ ∅ conditions⊆ ⊆ are satisﬁed.⊆ ∈R W { } # Now assume that for some natural number n, τ , τ , τ τ n have been U V W | |≤ constructed so that all four conditions (i) - (iv) hold, and let λ [κ]n+1.Foreach ∈ τ(λput τ λ τ λ Fτ = Vτ − xκ− . × × Let F = Fτ : τ ( λ . { } [ Clearly each Fτ is a compact subset of .Foreachτ(λwe have τ n, hence | |≤ the inductive assumption implies (see (i), (iva) and (iii)) that Fτ τ .Thus ⊆ W

# S (1) F is a compact subset of and F τ : τ ( λ . ⊆ W [ n[ o Choose a family # of open boxes in ,sothat: Wλ (2) W # R such that W R, ∀ ∈Wλ ∃ ∈R ⊆

(3) # is disjoint from F, Wλ [

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and (see (i) and (1) above)

# # λ κ λ (4) λ = τ : τ ( λ is an irreducible open covering of x − . W Wλ ∪ W × [ Clearly # is ﬁnite, hence we can assume without loss of generality that for Wλ some open box neighborhood Uλ of x with

(5) Uλ Vτ : τ ( λ , ⊆ { } \ we have

# λ λ (6) for all W ,Wκ− =Uκ− . ∈Wλ λ

Finally we choose any box neighborhood Vλ of x,sothat

(7) Vλ Uλ. ⊆ Now (4) implies (i), (2) implies (ii), (5) and (7) give us (iii), while (6) implies (iva) and (3) implies (ivb). Now Lemma 3.1 enables us to prove the following result which is probably in- teresting in its own.

Lemma 3.2. Let κ be an infinite cardinal and let Xα be a compact Hausdorff space for each α κ.Letx =α κXαand let O be an open neighborhood of σ(x). There exists∈ an open neighborhood∈ ∈ P of σ(x) such that P O. ⊆ ProofofLemma3.2. = O is a 1-element open cover of σ(x). Let , and be families of openR boxes{ constructed} for as in the proof of LemmaU 3.1.V Put WP = . Clearly P is an open neighborhoodR of σ(x)in . W ClaimSS 3.2.1. P O. ⊆ Proofoftheclaim. Let y O. We will find an open neighborhood of y disjoint ∈ \ from P. We put τ 1 = ∅ and construct by induction on n subsets θn n<∆ and − h i τn n<∆ of κ andopenboxneighborhoods Tn n<∆ of y,forsome∆ ω+1,until h i h i ∈ the first (if any) infinite θn is found. For n = 0 we put:

θ0 = γ κ : y(γ) / U (γ) , (1) ∈ ∈ ∅ τ0 = θn0, o

and inductively if θn and τn are deﬁned for some n ω,andifθn is ﬁnite, we put: ∈

θn+1 = γ/τn:y(γ)/Uτ (γ) , (2) ∈ ∈ n τn+1 = τnn θn+1. o ∪ Notice that

(3) n<∆τn = θm and θn τn 1 = ∅. ∀ ∩ − m n [≤

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Note also that

(4) n<∆θn =∅. ∀ 6 # τn 1 τn 1 To see (4) choose W τn 1 so that y − W − .Sincey/Wand (compare ∈W − ∈ ∈ (iva) in Lemma 3.1) α/τn 1 W(α)=Uτn 1(α), there exists γ/τn 1such that ∀ ∈ − − ∈ − y(γ) / Uτn 1 (γ). Clearly γ θn. Since∈ we− stop our inductive∈ construction at the ﬁnite stage n only when we get an inﬁnite θn, (4) implies that

(5) Θ = θn is infinite . n<∆ [ Now, for each n<∆, that is, for each n for which θn and τn are defined, we define Tn as follows: α κ ∀ ∈

Tn(α)=Xα Uτm 1 if α θm for some m n, (6) \ − ∈ ≤ Tn(α)=Xα if α κ θm. ∈ \ m n [≤

Subclaim. T = Tn : n<∆ is an open box neighborhood of y disjoint from P . { } Proof of the subclaim.T By (6) we get: n γ θn T (γ)=Tn(γ), (7) ∀ ∀ ∈ γ/ΘT(γ)=Xγ, ∀ ∈ and since n<∆y Tn,Tis an open box neighborhood of y. Next ∀ ∈ <ω # (8) λ [κ] n ω W if W Tn = ∅, then τn λ. ∀ ∈ ∀ ∈ ∀ ∈Wλ ∈W ∩ 6 ⊆ We prove (8) by induction on n.Forn= 0 and each α κ we have (see (iii),(iva) in Lemma 3.1 and (1)) ∈

α/λ W ( α )=Uλ(α) U (α), ∈ −→ ⊆ ∅ α θ0 W ( α ) U ( α ) = ∅ . ∈ −→ \ ∅ 6 Therefore α θ0 α λ and τ0 = θ0 λ. ∈ −→ ∈ ⊆ # Now assume that (8) holds for some n ω and let W W meet Tn+1. W ∈ ∈ λ ∩ Tn+1 = ∅ W T n = ∅ and by the induction hypothesis we get τn λ.For 6 −→ ∩ 6 ⊆ each α κ we have (see (iii),(iva) in Lemma 3.1 and (2)) ∈ α/λ W ( α )=Uλ(α) Uτ (α), ∈ −→ ⊆ n

α θn+1 W ( α ) U τ n ( α ) = ∅ . ∈ −→ \ 6 Therefore α θn+1 α λ and τn+1 = τn θn+1 λ. This concludes the proof of (8). ∈ −→ ∈ ∪ ⊆ # By (8) if W meets T = Tn : n<∆ ,thenΘ λ. It is clearly ∈Wλ ∈W { } ⊆ impossible since λ is ﬁnite and Θ is inﬁnite (see (5)). Hence T = ∅ and T ∩ W the proof of the subclaim is completed. SS By the subclaim y/P, and since y was an arbitrary point in O we get P O. This completes∈ the proof of the claim and the proof of the lemma.\ ⊆ Using the last lemma and Theorem 2.1 we easily get the theorem mentioned in the abstract.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 314 PETER NYIKOS AND LESZEK PIA¸ TKIEWICZ

Theorem 3.3. Let κ be an infinite cardinal and let Xα : α κ be a family of { ∈ } compact Hausdorff spaces. Let x = α Xα be a fixed point and let be a ∈ ∈κ R family of open sets in which covers σ(x). There exists an open locally finite in family which strictly refines and covers σ(x). S R Proof of Theorem 3.3. Let U be an open neighborhood of σ(x)andlet be a locally finite in U family of open sets which strictly refines and coversBU (see Theorem 2.1). By Lemma 3.2 we can choose an open neighborhoodR V of σ(x)so that V U.Put = B V:B . Clearly is a family of open subsets of which strictly⊆ refinesS {and∩ covers∈B}σ(x). is locallyS finite in U and is disjoint R S S from V , hence is locally finite in U ( V )=. \ S ∪ \ S Question 3.4. Does Theorem 3.3(equivalently Lemma 3.2) remain true if one drops the assumption of compactness of Xα’s and assumes only that each finite product of Xα’s is paracompact Hausdorff ? In other words, is the natural common generalization of Theorems 2.1 and 3.3 valid ? ω Recall that the Σ-product is the subspace of 1 (ω + 1) defined as follows:

ω1 x (ω +1): α ω1 :x(α)=ω is countable . { ∈ { ∈ 6 } } As stated in the introduction the Σ-product is consistently nonnormal. The following question remains open: Question 3.5. Is the Σ-product nonnormal in ZFC?

References

[vD] E. K. van Douwen, The box product of countably many metrizable spaces need not be normal, Fund. Math. 88 (1975), 127–132. MR 52:6640 [E] R. Engelking, General topology, PWN, Warszawa, 1977. MR 58:18316b [K] C.J.Knight,Box topologies, Quart. J. Math. Oxford 15 (1964), 41–54. MR 28:3398 [L] L. B. Lawrence, Failure of normality in the box product of uncountably many real lines, preprint. CMP 95:04 [vM] J. van Mill, Collected papers of Eric K. van Douwen,preprint. [R] M. E. Rudin, Lectures on set-theoretic topology, CBMS Regional Conf. Ser. in Math., vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR 51:4128 [W] S. W. Williams, Paracompact sets in box products,preprint.

Department of Mathematics, University of South Carolina, Columbia, South Car- olina 29208 E-mail address: [email protected]

Department of Mathematics and Computer Science, Pembroke State University, Pem- broke, North Carolina 28372 E-mail address: [email protected]

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