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Metrisable and Discrete Special Resolutions CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Topology and its Applications 122 (2002) 605–615 Metrisable and discrete special resolutions Kerry Richardson a,∗, Stephen Watson b,1 a Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand b Department of Mathematics and Statistics, 4700 Keele Street, York University, Toronto, Canada Received 10 November 2000; received in revised form 8 March 2001 Abstract In this article we give characterisations for a topological space constructed by special resolutions to be metrisable and discrete, respectively. We show how nice topological properties can be completely destroyed by resolutions. 2002 Elsevier Science B.V. All rights reserved. AMS classification: 54B99 Keywords: Special resolution; Fully closed maps 1. Introduction The method was introduced by Fedorcukˇ [3] and developed by Watson [10]. A lot of results of great importance have been obtained using this method. Among them; an example of a compact space X with different dimensions (covering dimension dimX = 2 and inductive dimensions satisfying 3 indxX 4foreveryx ∈ X) in [3], consistent examples of non-metrisable compact spaces with hereditarily separable finite powers in [7], and a CH example of a regular space X with a countable network and different dimensions (dimX = 1andindX = IndX 2) in [1]. \{ }→ Definition 1. Let X and Yx be non-empty topological spaces and fx : X x Yx be a continuous function for each x ∈ X. The point set is Z = {{x}×Yx : x ∈ X}. * Corresponding author. Partially supported by: The University of Auckland Postgraduate Research Fund; the Maori Education Trust; the New Zealand Mathematical Society; and a Marsden Fund Award, UOA611, from the Royal Society of New Zealand. E-mail addresses: [email protected] (K. Richardson), [email protected] (S. Watson). 1 Supported by the Natural Sciences and Engineering Research Council of Canada. 0166-8641/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0166-8641(01)00185-7 606 K. Richardson, S. Watson / Topology and its Applications 122 (2002) 605–615 Define projections π : Z → X by π(x,y)= x and σx : Z → Yx by y if x = x, σx x ,y = fx (x ) if x = x. Take any z ∈ Z.Letπ(z) = x ∈ U an open subset of X and σπ(z)(z) ∈ W, an open subset of Yx . We define a typical element of a local base for Z at z to be ⊗ = −1 ∩ −1 U W π (U) σx (W). We say U ⊗ W is based on the point z (or x when it is more convenient to talk about points in X). Conventions. Conventions, assumptions and terminology used throughout this paper are: (1) X is always a non-empty T1 topological space, (2) {x}×Yx is identified with Yx , (3) X is referred to as a “base” space and Yx as a “fibre”, (4) “resolve X” means apply the process in Definition 1 to X, (5) a “resolution” or “Z” is the topological space one obtains after applying the above process to X. Useful Facts. Extensive use is made throughout this paper of the following observations: (1) the topology on {x}×Yx as a subspace of Z is the same as the topology on Yx via a natural embedding, { }× (2) x Yx is a closed subspace of Z, − { }× ∪ {{ }× ∈ ∩ 1 } (3) the set ( x W) x Yx : x U fx (W) can also be given as the definition of U ⊗ W, −1 = ⊗ −1 = ⊗ (4) π and σx are continuous since σx (W) X W, π (U) U Yx . We will use both forms of U ⊗ W interchangeably. Let Λ denote the set {x ∈ X: |Yx| > 1}. For points in X \ Λ it is assumed that they are resolved into a one point space by constant maps. A subset of X is called Fσ -discrete if it can be represented as a union of countably many closed discrete subsets of X;soan Fσ -discrete set need not be discrete. 2. Metrisability 2.1. σ -locally finite families Proposition 2. Assume X is a metrisable space and Λ is a Fσ -discrete subset of X and each Yx has a σ -locally finite base. Then Z has a σ -locally finite base. Proof. The proof follows that of the corresponding proposition for general resolutions (a generalisation of special resolutions) in [8], except that here we do not have to take care of K. Richardson, S. Watson / Topology and its Applications 122 (2002) 605–615 607 the variable J in the special resolution case. So now the locally finite family we construct is Lij k = ⊗ ∈ Yi ∈ k B(x,j) W: W x ,x C , where all symbols mean what they did for the corresponding proposition in [8]. The family Zik ={π−1(U \ Ck): U ∈ X i } is also locally finite and (Lij k ∪ Zik) is a basis for the special resolution topology. ✷ Corollary 3. Assume X is a metrisable topological space and each topological space Yx has a σ -locally finite base. If Λ is a closed discrete subset of X then Z has a σ -locally finite base. Corollary 4. Assume X is a metrisable space and each Yx has a σ -locally finite base. If Λ is a countable subset of X then Z has a σ -locally finite base. 2.2. Non-metrisable special resolutions Definition 5. If a topological space (X, τ) is resolved at each x ∈ X into a space (Yx ,τx) by continuous mappings fx : X \{x}→Yx , then the boundary of Yx , denoted by Bd(Yx) is −1 Bd(Yx) = y ∈ Yx : ∀U ∈ τ, x ∈ U, (x,y) ∈ cl π U \{x} . The interior of Yx denoted by intYx is the set Yx \ Bd(Yx). This definition says that y ∈ Yx is a boundary point if and only if for every open sets ∈ ∈ ∈ ∈ ∩ −1 =∅ U τ and W τx such that x U, y W we have U fx (W) . Lemma 6. If each Bd(Yx) is compact then π is a closed map. Proof. The proof is a variation of the Kuratowski Theorem [2, Theorem 3.1.16]. Take any closed subset F ⊂ Z. We will show that π(F) is closed by showing that X \ π(F) is open in X. Choose any x ∈ X \ π(F). It suffices to find an open U ⊂ X with x ∈ U such that −1 π (U) ∩ F =∅.Asx ∈ X \ π(F) we have {x}×Yx ⊂ Z \ F .Foreveryy ∈ Yx the point (x, y) has a neighbourhood of the form Uy ⊗ Wy ⊂ Z \ F where each Uy is an open set in X containing x.SinceBd(Yx) is compact there exists a finite set {y1,...,yk} such that { }× ⊂ ⊗ x Bd(Yx ) (Uyi Wyi ). ik = = = \ = ∪ Let U ik Uyi ,W ik Wi and W Yx W. Note that Yx W W . Since, in ∩ −1 ∩ =∅ ⊗ ∩ =∅ particular, Uyi fx (Wyi ) π(F) for every i k therefore U W F .Also U ⊗ (int Yx ) ={x}×int Yx is clearly disjoint from F . Because the union of these two sets, −1 each disjoint from F ,isU ⊗ Yx = π (U) we are done. ✷ 608 K. Richardson, S. Watson / Topology and its Applications 122 (2002) 605–615 Proposition 7. A resolution of a topological space X is always non-metrisable if Bd(Yx) is compact for each x and Λ is not Fσ -discrete. Proof. Suppose Z is metrisable; choose a metric d on Z. For each x ∈ Λ we choose an ∗ integer nx ∈ ω as follows. Choose distinct z ,z ∈{x}×Yx and let nx ∈ ω be such that ∗ 1 <d(z ,z).DefineCn ={x ∈ Λ: n = n} for n ∈ ω. Since Λ is not F -discrete, then nx x σ Cn is either not discrete or is not closed for some n ∈ ω. Thus, Cn has an accumulation n point x,i.e.,x ∈ clX(C ). By Lemma 6 π is closed; and Z is assumed metrisable. By the n Hanai–Morita–Stone Theorem X is metrisable. Thus, there is a sequence xi in C which converges to x such that xi = x for every i ∈ ω. Since Bd(Yx ) is a compact subset of a metric space the sequence fx (xi) has a subsequence converging to some y ∈ Yx . By passing to this subsequence if necessary, we may assume that fx (xi) converges to y. Let U ⊗W be any basic open neighbourhood of z = (x, y); without loss of generality we ⊗ ∈ ∈ can assume U W is based on z since we have a local base. Then x U and y W so there − − ∈ ∈ ∩ 1 ∈ {{ }× ∈ ∩ 1 }⊂ is an m ω such that xi U fx (W) and (xi,ηi ) x Yx : x U fx (W) U ⊗ W for every i m. So the sequence (xi,ηi ) in the resolution space converges to z, ∈ whatever ηi Yxi may be. We now show that this fact is violated. ∗ −1 n Consider two distinct sequences zi and zi in π (C ) that converge to z such that ∗ = ∗ = = ∗ 1 ∗ z z and π(z ) π(z ) xi and z and z are as in the definition of nxi ; <d(z ,z ) i i i i i i nxi i i for every i ∈ ω.LetBε(z) be the open ball in Z of radius ε and centred at the limit point z. 1 ∗ ∗ + ∈ We have n <d(zi ,zi ) d(zi ,z) d(z,zi)<εfor all but finitely many i ω. We obtain = 1 ✷ a contradiction by choosing ε n . A space is ω1-compact if there are no uncountable closed discrete subsets of the space. Corollary 8.
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