summary:Through the study of frame congruences, new characterizations of the paracompactness of frames are obtained

Chen, Xiangdong

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeXiangdong ChenOn the paracompactness of framesCommentationes Mathematicae Universitatis Carolinae, Vol. 33 (1992), No. 3, 485--491Persistent URL: http://dml.cz/dmlcz/118516Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1992Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 33,3 (1992)485–491 485On the paracompactness of framesChen, XiangdongAbstract. Through the study of frame congruences, new characterizations of the paracom-pactness of frames are obtained.Keywords: paracompact, cover, envelope, congruenceClassification: 54D20, 54J05, 06B10, 06D99There are some frame properties whose corresponding topological properties haveimportant characterizations in terms of subsets that are not necessarily open. Forinstance, local connectedness and paracompactness are such properties. To getframe counterparts of these valuable characterizations, we turn to frame congru-ences, which provide sufficient tools for us to translate directly some topologicalproperties pertaining to general subsets, especially, closed subsets of spaces. Ad-vantages of such considerations have been shown through the study of the localconnectedness of frames in [2].It is well known that the paracompactness of spaces can be characterized byemploying one of the following refinements: (1) locally finite; (2) cushioned;(3) closure-preserving; (4) σ-locally finite open; (5) σ-closure preserving open;and (6) σ-cushioned open. Through the study of congruences, and applying theresults of [3], we obtain the frame versions of the above classical characterizations,and thus extend the related results of [3], [6] and [7]. The topological intuitionbehind our arguments concerning congruences can be easily traced by knowing thecorrespondence between congruences and subspaces.Throughout this paper, L always represents a frame.1. Basic facts.For general background of frames, we refer to [1] and [4].1.1. For a frame L, its top (bottom) element will be denoted by e (0). For a ∈ L, itspseudocomplement is denoted by a∗ and is given by a∗ =∨{x ∈ L | x ∧ a = 0}.For any X ⊆ L, we use X∗ to denote the set {x∗ | x ∈ X}.The relation ≺ on L is defined such that b ≺ a if and only if b∗ ∨ a = e. A frameL is called regular if a =∨{x ∈ L | x ≺ a} for each a ∈ L.A frame L is called normal if for a1, a2 ∈ L such that a1 ∨ a2 = e there areb1, b2 such that a1 ∨ b1 = a2 ∨ b2 = e and b1 ∧ b2 = 0.This paper was written while I was a postdoctoral research assistant of Professor B. Ba-naschewski at McMaster University. The financial support is gratefully acknowledged486 Chen, Xiangdong1.2. The lattice of frame congruences on L under set inclusion is a frame, denotedby CL. The top and bottom of CL are denoted by ∇ and △. We reserve smallGreek letters to denote congruences.The following properties, extracted from [1], are very useful:(1.2.1) For any a ∈ L, ∇a = {(x, y) | x ∨ a = y ∨ a}, called closed, is the leastcongruence containing (0, a); △a = {(x, y) | x∧a = y∧a}, called open, is the leastcongruence containing (e, a).(1.2.2) Each ∇a is complemented in CL with complement △a.(1.2.3) For any a ∈ L, the map a  ∇a is a frame embedding L −→ CL, whereas,the map a △a is a dual poset embedding L −→ CL taking finitary ∧ to finitary∨ and arbitrary∨to arbitrary∧.(1.2.4) For any θ ∈ CL and any a ∈ L,∇a ∨ θ = {(x, y) | (x ∨ a, y ∨ a) ∈ θ} and △a ∨ θ = {(x, y) | (x ∧ a, y ∧ a) ∈ θ}.1.3. For any θ ∈ CL, the closure of θ, denoted by θ, is defined to be the ∇a wherea is the largest element such that (0, a) ∈ θ. It is obvious that, for any a ∈ L,△a = ∇a∗ .2. Covers and envelopes.2.1. A subset A of L is said to be a cover of L if∨A = e. For any two covers Aand B, A is called a refinement of B if for each a ∈ A there exist b ∈ B such thata ≤ b. Notation A ≤ B.A subset T of L is called locally finite if there exists a cover A such that, foreach a ∈ A, {t ∈ T | t ∧ a 6= 0} is finite. Such a cover A is said to finitize T .A subset U of L is said to be an envelope if x =∧{x ∨ u | u ∈ U} for eachx ∈ L. (Such a U is also called a closed covering of L by [3]). An envelope U iscalled a corefinement of an envelope V if for each u ∈ U there exist v ∈ V suchthat v ≤ u.A subset T of L is called conservative if for each S ⊆ T and each x ∈ L,x ∨∧S =∧{x ∨ t | t ∈ S}.An envelope U is called a dual-refinement of a cover A if for each u ∈ U thereexist a ∈ A such that u ∨ a = e.2.2. Lemma. For any T ⊆ L and any θ ∈ CL,θ ∨∧{△t | t ∈ T } =∧{θ ∧△t | t ∈ T }.Therefore, O(T ) = {△t | t ∈ T } is conservative in CL.Proof: By applying (1.2.4). 2.3. Lemma. For any T ⊆ L,∧{∇t | t ∈ T } = ∇VT if and only if ∧ T ∨ x =∧{t ∨ x | t ∈ T } for all x ∈ L.Proof: (⇐=) Take any (x, y) ∈∧{∇t | t ∈ T }, x ∨ t = y ∨ t for all t ∈ T , thenx ∨∧T = y ∨∧T , which means (x, y) ∈ ∇VT .On the paracompactness of frames 487(=⇒) Consider (∧T ∨ x,∧{t∨x | t ∈ T }). For each t ∈ T , x∨ t =∧T ∨ x∨ t =∧{t ∨ x | t ∈ T } ∨ t, hence (x,∧{t ∨ x | t ∈ T }) ∈∧{∇t | t ∈ T } = ∇VT , whichmeans∧T ∨ x =∧{t ∨ x | t ∈ T } for all x ∈ L. 2.4. A set Θ of congruences of L is called locally cofinite if there is a cover A ofL such that, for each a ∈ A, ∇a ≤ θ for all but finitely many θ ∈ Θ. Such a coverA is said to co-finitize Θ.The following are trivial observations:(2.4.1) Θ ⊆ CL is locally cofinite if and only if Θ̄ = {θ̄ | θ ∈ Θ} is locally cofinite.(2.4.2) A subset T of L is locally finite if and only if O(T ) = {△t | t ∈ T } is locallycofinite.2.5. Lemma. For any T ⊆ L, if C(T ) = {∇t | t ∈ T } is locally cofinite, then(1) C(T ) is conservative in CL.(2)∧{∇t | t ∈ T } = ∇VT .(3)∨{△t | t ∈ T } = △VT .Proof: Let A be a cover of L co-finitizing C(T ). Take an arbitrary element a ∈ A.Let T0 = {t ∈ T | a 6≤ t}, T0 is finite and a ≤∧(T − T0).For any (x, y) ∈∧{θ ∨∇t | t ∈ T }, (x ∨ t, y ∨ t) ∈ θ for all t ∈ T . Then∧T ∨ (x ∧ a) =∧(T − T0) ∧∧{t ∨ (x ∧ a) | t ∈ T0}≡θ∧(T − T0) ∧∧{t ∨ (y ∧ a) | t ∈ T0} =∧T ∨ (y ∧ a).Therefore,∧T ∨x ≡θ∧T ∨ y, that is, (x, y) ∈ θ∨∇VT ≤ θ∨∧{∇t | t ∈ T }. Thisproves (1).(2) and (3) are easy now. 2.6. Lemma. If Θ ⊆ CL is locally cofinite, then∧{θ | θ ∈ Θ} =∧{θ̄ | θ ∈| Θ}.Proof: By (2.4.1) and (2.5),∧{θ̄ | θ ∈| Θ} is closed. 2.7. Proposition. For any locally finite T ⊆ L, T ∗ = {t∗ | t ∈ T } is conservative.Proof: That O(T ) = {△t | t ∈ T } is locally cofinite implies that O(T ) = {∇t∗ |t ∈ T } is locally cofinite. By (2.5) and (2.3),x ∨∧{t∗ | t ∈ T } =∧{x ∨ t∗ | t ∈ T } for all x ∈ L.2.8. For two subsets T, S of L, we use T ≺ S to mean that there is a functionf : T −→ S such that, for each subset X ⊆ T ,∨X ≺∨f [X ]. Recall that T is saidto be cushioned in S (see [3]) if there is a function f : T −→ S such that, for eachsubset X ⊆ T ,∧X ∨∨f [X ] = e. Therefore, T ≺ S means that T ∗ is cushionedin S.488 Chen, Xiangdong2.9. Proposition. For T, S ⊆ L, if for each t ∈ T there exist s ∈ S such thatt ≺ s, and T ∗ is conservative, then T ≺ S.Proof: From the given condition, we obtain a function f : T −→ S such thatt ≺ f(t) for all t ∈ T . Then for any subset X of T ,∨f [X ] ∨ (∨X)∗ =∨f [X ] ∨∧{t∗ | t ∈ X} =∧{∨f [X ] ∨ t∗ | t ∈ X} = e,that is,∨X ≺∨f [X ]. Hence T ≺ S. 2.10. Lemma. For every countable cover {an | n = 1, 2, · · · } of L, there is a locallycofinite {θn | n = 1, 2, · · · } of CL such that∞∧n=1θn = △ and △an ≤ θn, n = 1, 2, · · · .Proof: Takeθ1 = △a1 , θn = △an ∨∨{∇ai | i = 1, . . . , n − 1},then {θn | n = 1, 2, · · · } is locally cofinite.Suppose x 6= y and (x, y) ∈∧{θn | n = 1, 2, · · · }. Suppose k is the smallestnumber such that (x, y) /∈ △ak . Then (x, y) ∈∧{△ai | i = 1, · · ·k − 1}. Then(x, y) ∈ (△ak ∨∨{∇ai | i = 1, . . . , k − 1}) ∧∧{△ai | i = 1, · · ·k − 1} ≤ △ak ,a contradiction. Hence∧{θn | n = 1, 2, · · · } = △. 3. Paracompactness.3.1. A frame is called paracompact if each cover has a locally finite refinement.3.2. For a cover A = {ai | i ∈ I} of L, we call(1) A shrinkable if there is a cover B = {bi | i ∈ I} such that bi ≺ ai for alli ∈ I (see [3]).(2) A strongly shrinkable if there is a cover B such that B ≺ A.(3) A σ-strongly shrinkable if A has a refinement B =⋃∞n=1Bn such thatBn ≺ A.Lemma. Every strongly shrinkable cover is shrinkable.Proof: Let A = {ai | i ∈ I} be a strongly shrinkable cover, then there is a coverB satisfying B ≺ A with a corresponding function f : B −→ A. For each i ∈ I, putbi =∨{b ∈ B | f(b) = ai}, which satisfies bi ≺ ai. On the paracompactness of frames 4893.3. Theorem. The following properties are equivalent:(i) L is paracompact and normal.(ii) Each cover of L has a conservative envelope as a dual-refinement.(iii) For each cover A of L, there is an envelope U being cushioned in A.(iv) For each cover A of L, there exists a Θ ⊆ CL satisfying∧Θ = △ anda function f : Θ −→ A such that for every subset Θ′ of Θ,∧{△f(θ) | θ ∈ Θ′} ≤∧{θ | θ ∈ Θ′}.(v) Every cover of L is σ-strongly shrinkable.Proof: The equivalences for (i), (ii) and (iii) have been proved by Dowker andPapert Strauss [3].(i) =⇒ (v). For each cover A of L, there is a locally finite refinement B of A.By [3, Proposition 1], B is shrinkable, so there is a cover C such that for each c ∈ Cthere exist b ∈ B such that c ≺ b. Again C has a locally finite refinement D, thenD ≺ B by (2.7) and (2.9), implies D ≺ A. Thus A is strongly shrinkable.(v) =⇒ (iv). Consider a cover A of L and a refinement B =⋃∞n=1Bn of A suchthat Bn ≺ A with corresponding functions fn : Bn −→ A for n = 1, 2, · · · .Putdn =∨Bn, n = 1, 2, · · · .By (2.10), there exists a locally cofinite {θn | n = 1, 2, · · · } of CL such that∧∞n=1 θn = △ and △dn ≤ θn for all n.PutΦ = {θn ∨△b | b ∈ Bn, n = 1, 2, · · · }.(1)∧Φ =∧∞n=1∧{θn∨△b | b ∈ Bn} =∧∞n=1(θn∨△dn) =∧∞n=1 θn = △, wherethe second equality holds since θ ∧△∨X =∧{θ ∨△x | x ∈ X} for any X ⊆ L andθ ∈ CL.(2) Define f : Φ −→ A by f(θn ∨ △b) = fn(b). For any subset Φ′ ⊆ Φ withΦ′ = {θn ∨△b | b ∈ B′n, n = 1, 2, · · · } where B′n ⊂ Bn,∧Φ′ =∧{θn ∨△WB′n | n = 1, 2, · · · }=∧{θn ∨△WB′n | n = 1, 2, · · · } by (2.6)≥∧{∇(WB′n)∗ | n = 1, 2, · · · }≥∧{△W fn[B′n] | n = 1, 2, · · · }=∧{△fn(b) | b ∈ B′n, n = 1, 2, · · · }.Thus Φ has the property required for Θ in (iv).(iv) =⇒ (iii). Since for every Θ′ ⊆ Θ,∧{θ | θ ∈ Θ′} =∧{θ̄ | θ ∈ Θ′},490 Chen, Xiangdongwe may choose Θ = {∇b | b ∈ B} consisting of closed congruences of L. Then B isan envelope of L by (2.3). Define g : B −→ A by g(b) = f(∇b). For every B′ ⊆ B,∧{△g(b) | b ∈ B′} ≤∧{∇b | b ∈ B′}implies∨g[B′] ∨∧B′ = e. Hence B is cushioned in A. 3.4. Remark. Recall a classical result (see [5]) that for a T1-spaceX , the followingproperties are equivalent:(i) X is paracompact.(ii) Every open covering X has a cushioned (not necessarily open) refinement.(iii) Every covering of X has a σ-cushioned open refinement.The equivalences of (i), (iv) and (v) of (3.3) can be regarded as the frame coun-terpart of this classical results.3.5. Theorem. For a regular frame L, the following are equivalent:(i) L is paracompact.(ii) Each cover of L has a σ-locally finite refinement.(iii) Each cover A of L has a refinement B =⋃∞n=1Bn such that B∗n is conser-vative for every n.(iv) For each A of L, O(A) = {△a | a ∈ A} has a locally cofinite corefinement Θ.Proof: (i) =⇒ (ii) =⇒ (iii) and (i) =⇒ (iv) are obvious.(iii) =⇒ (i). Let A be a cover of L. Put C = {c | c ≺ a for some a ∈ A},which is also a cover of L. Then C has a refinement B =⋃∞n=1Bn such that B∗nis conservative. By (3.5), Bn ≺ A for all n. Therefore L is paracompact by (v)of (3.3).(iv) =⇒ (i). Let A be a cover of L. Put C = {c | c ≺ a for some a ∈ A}. Byassumption, O(C) has a locally cofinite corefinement Θ. Then for each θ ∈ Θ thereexist c ∈ C and a ∈ A such that △a ≤ ∇c∗ ≤ △c ≤ θ, implying △a ≤ θ̄. Thiscorrespondence gives a function f : Θ −→ A such that for every Θ′ ⊆ Θ,∧{△f(θ) | θ ∈ Θ′} ≤∧{θ̄ | θ ∈ Θ′} =∧{θ | θ ∈ Θ′}.Therefore L is paracompact by (iv) of (3.3). 3.6. Proposition. Every complete Boolean algebra is paracompact.A proof of this has been given by Sun [7]. Here we can give a simple proof basedon (3.5).Proof: For every cover A of a complete Boolean algebra, A∗ is conservative. Thusthe claim holds by (iii) of (3.5). On the paracompactness of frames 4913.7. Proposition. If L is paracompact and regular, then L/θ is paracompactwhenever θ has a form of∧{∇tn | n = 1, 2, · · · }.Proof: Let q : L −→ L/θ be the quotient map. For each cover A of L/θ, we canfind a subset S of L such that q[S] = A, which implies(∨S, e) ∈ θ =∧{∇tn | n = 1, 2, · · · },that is,∨S ∨ tn = e for n = 1, 2, · · · . Then for each n there is a locally finiterefinement Cn of S ∪ {tn}. PutBn = {x ∈ Cn | x  tn},then∨Bn∨ tn = e. Thus (∨(⋃∞n=1Bn), e) ∈ θ, which means⋃∞n=1 q[Bn] is a coverof L/θ. Moreover each q[Bn] is locally finite and⋃∞n=1 q[Bn] ≤ q[S] = A. Hence Ahas a σ-locally finite refinement. By (3.5), we conclude that L/θ is paracompact.References[1] Banaschewski B., Lectures on Frames, University of Cape Town, 1988 (unpublished manu-script).[2] Chen X., On the local connectedness of frames, J. Pure Appl. Algebra 79 (1992), 35–43.[3] Dowker C.H., Papert D., Paracompact frames and closed maps, Symp. Math. 16 (1975),93–116.[4] Johnstone P.T., Stone Space, Cambridge University Press, 1982.[5] Nagata J., Modern General Topology, N.Y., Wiley, 1968.[6] Pultr A., Úlehla J., Notes on characterization of paracompact frames, Commentationes Math.Univ. Carolinae 30 (1989), 377–384.[7] Sun Shu-Hao, On paracompact locales and metric locales, Commentationes Math. Univ. Ca-rolinae 30 (1989), 101–107.Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario,Canada L8S 4K1(Received March 27, 1992)

On the paracompactness of frames

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On the paracompactness of frames

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Institute of Mathematics AS CR, v. v. i.