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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<m Xn are paracompact, then for each element x of the box product n ωXn the σ-product σ(x)= ∈Q y n ωXn : n ω:x(n)=y(n) is finite is paracompact. He asked whether{ ∈ this∈ result{ remains∈ true if6 one considers} uncountable} families of spaces. In this paper we prove in particular the following result: 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 304 PETER NYIKOS AND LESZEK PIA¸ TKIEWICZ 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PARACOMPACT SUBSPACES IN THE BOX PRODUCT TOPOLOGY 305 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 para- compact. 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 neighbor- hoods 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) , ⊆ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 306 PETER NYIKOS AND LESZEK PIA¸ TKIEWICZ 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 .
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