For a given locally compact Hausdorff space we introduce a Boolean algebra structure on the family of all its one-point local compactifications

Polański, Artur

Journal of Mathematics and Applications

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Boolean Algebra of One-Point Local Compactifications

Jagiellonian Univeristy Repository

J o u r n a l ofMathematicsand ApplicationsJMA No 43, pp 113-121 (2020)COPYRIGHT © by Publishing House of Rzeszów University of TechnologyP.O. Box 85, 35-959 Rzeszów, PolandBoolean Algebra of One-Point LocalCompactificationsArtur PolańskiAbstract: For a given locally compact Hausdorff space we intro-duce a Boolean algebra structure on the family of all its one-point localcompactifications.AMS Subject Classification: 54D45, 03G05.Keywords and Phrases: Local; Compactification; Boolean; Algebra; Ends.1. IntroductionEvery locally compact, noncompact Hausdorff space X has a well known one-pointcompactification (Alexandroff compactification, [1]). In this paper we consider theset B(X) of all one-point local compactifications of X up to an equivalence. Weprove that B(X) is a partially ordered set such that the order 6 induces a Booleanalgebra. Moreover, the elements 0 and 1 of B(X) are respectively X and ωX. Thenwe focus on describing the algebra we get using topological notions and convergenceand we provide examples by computing the algebra in some special cases. We alsonote the connection with the topic of ends of manifolds (see [2, pages 110-118]), asfor a noncompact, connected, second countable manifold L with n ends, n < ∞, wehave |B(L)| = 2n.2. Notation and terminology• Throughout the paper, ZFC is assumed.• Given a locally compact Hausdorff space X we denote by ωX a one-point com-pactification of X if X is not compact and X otherwise,114 A. Polański• a clopen set is a set that is both closed and open,• if Y is a one-point local compactification different from X, the unique point ofY \X will be denoted by ∞Y ,• a filter F of open sets in a topological space X is a non-empty family of setsopen in X such that ∅ /∈ F and, for all V1, V2 ∈ F and an open V ⊂ X we haveV1 ∩ V2 ∈ F ⇒ V ∈ F .3. Main resultsDefinition 1. If X is a locally compact Hausdorff space, we call (Y, f) an at most one-point local compactification of X iff Y is a locally compact Hausdorff and f : X → Yis a homeomorphic embedding such that f(X) is dense in Y and |Y \ f(X)| 6 1. If(Y, f) is an at most one point local compactification of X and |Y \ f(X)| = 1, we call(Y, f) a one-point local compactiication of X.For simplicity, we say that Y is a/an (at most) one-point local compactification of Xiff (Y, idX) is a/an (at most) one-point local compactification of X.Definition 2. Let X be a locally compact Hausdorff space, (Y1, f1) and (Y2, f2) itsat most one-point local compactifications. We will write (Y1, f1) 6 (Y2, f2) (or, forsimplicity, Y1 6 Y2) iff one of the following conditions apply:• f1(X) = Y1• Y1 = f1(X) ∪ {∞Y1}, Y2 = f2(X) ∪ {∞Y2} and the functionY1 3 x 7→{f2(f−11 (x)), x ∈ f1(X)∞Y2 , x =∞Y1∈ Y2is continuous.Note that 6 is reflexive and transitive, with 0 = X and 1 = ωX. We can definean equivalence relation ≡ by(Y1, f1) ≡ (Y2, f2) iff (Y1, f1) 6 (Y2, f2) and (Y2, f2) 6 (Y1, f1),or, for simplicity,Y1 ≡ Y2 iff Y1 6 Y2 and Y2 6 Y1.We also defineB(X) := {Y—one-point local compactification of X}/≡.From now on instead of an equivalence class of Y in B(X) we will just write Y .We are now ready to state the first result where we will prove that B(X) orderedby 6 is a Boolean algebra, by showing that it is in fact order isomorphic to a muchsimpler one.Boolean Algebra of One-Point Local Compactifications 115Theorem 1. Given a locally compact Hausdorff space X, B(X) is a partially orderedspace with a lattice such that the order 6 induces a Boolean algebra, i.e., for Y1, Y2one-point local compactifications of X:• Y1 ∨ Y2 = sup6{Y1, Y2},• Y1 ∧ Y2 = inf6{Y1, Y2},• 0 = X,• 1 = ωX,• for any space Y ∈ B(X) there exists a unique space \Y ∈ B(X) : Y ∧ \Y = 0,Y ∨ \Y = 1.In particular, 0 = 1 iff X is compact.Proof. First consider βX, a Čech–Stone compactification of X. We define A(X) :={F ⊂ βX \ X : F clopen in βX \ X} (note that βX \ X is compact). A(X) withstandard set operations is a Boolean algebra. We will show an isomorphism betweenB(X) and A(X), proving that B(X) is also a Boolean algebra.To this end, we will define f : B(X) → A(X). If X is compact, both B(X)and A(X) are trivial, therefore assume that X is not compact. Consider a clopen inβX \X set F such that ∅ 6= F 6= βX \X. We can now identify F and (βX \X) \ Fwith points, getting a compact space X ∪ {{F}} ∪ {{(βX \ X) \ F}}. Its subspaceX ∪ {{F}} is then a one-point local compactification of X. Conversely, for any one-point local compactification Y of X there exists a unique clopen in βX \ X set FYsuch that Y is equivalent with X ∪ {{FY }} (from the universal property of βX).We define f(X) = ∅ and for every one-point local compactification Y of X we putf(Y ) = FY , where Y is the unique clopen in βX \ X set such that Y is equivalentto X ∪{{FY }}. It can be easily seen that for one-point local compactifications Y1, Y2of X we have Y1 6 Y2 iff FY1 ⊂ FY2 , so f preserves the partial order and is indeedan isomorphism. Furthermore, for one-point local compactifications Y1, Y2 of X wehave:1. Y1 ∨ Y2 = X ∪ {{FY1 ∪ FY2}}.2. Y1 ∧ Y2 = X ∪ {{FY1 ∩ FY2}} if FY1 ∩ FY2 6= ∅ and Y1 ∧ Y2 = X otherwise.3. \Y = X ∪ {{(βX \X) \ FY }} for ∅ 6= FY 6= βX \X.Remark 1. The proof of Theorem 1 shows that B(X) is isomorphic (as a Booleanalgebra) to the algebra of all clopen subsets of the remainder βX \X of X. One easilyconcludes that the Stone space of B(X) is homeomorphic to the space of all connectedcomponents of βX \X (that is, the space obtained from βX \X by identifying pointsthat lie in a common connected component).116 A. PolańskiNow that we know that B(X) is a Boolean algebra, we will focus on describing itwithout using A(X). If we add a point {∞Y } to a locally compact Hausdorff space Xto get its one-point local compactification Y , we only need to know the neighborhoodbasis at {∞Y } to know its topology. To this end, let us introduce the followingcharacterization. For simplicity, we will also use one more definition.Definition 3. Let X be a locally compact Hausdorff space, Y its one-point localcompactification. Thenτ(Y ) := {U\{∞Y } : U open neighborhood of ∞Y in Y }.τ(Y ) uniquely determines Y 6= X, Y ∈ B(X).Proposition 1. Let X be a locally compact Hausdorff space, Y1, Y2 ∈ B(X), Y1, Y2 6=0, Y1, Y2 6= 1.1. τ(Y1 ∧ Y2) = {U1 ∩ U2 : U1 ∈ τ(Y1), U2 ∈ τ(Y2)}, provided that the sets U1 ∩ U2are nonempty for all U1 ∈ τ(Y1), U2 ∈ τ(Y2) and Y1 ∧ Y2 = 0 otherwise.2. τ(Y1 ∨ Y2) = {U1 ∪ U2 : U1 ∈ τ(Y1), U2 ∈ τ(Y2)} = τ(Y1) ∩ τ(Y2).3. τ(\Y1) = {X\F : F ⊂ X, for any U ∈ τ(Y1) F\U compact}.Or, in terms of convergence:(a) A net (xγ) ⊂ X in Y1 ∧ Y2 is convergent to ∞Y1∧Y2 iff (xγ) is convergent to∞Y1 in Y1 and to ∞Y2 in Y2, and Y1 ∧ Y2 = 0 if there is no such net.(b) A net (xγ) ⊂ X in Y1 ∨ Y2 is convergent to ∞Y1∨Y2 iff every subnet of (xγ) hasa subnet convergent to ∞Y1 in Y1 or to ∞Y2 in Y2.(c) A net (xγ) ⊂ X in \Y1 is convergent to ∞\Y1 iff (xγ) has no convergent subnetsin Y1.Proof. Again, let βX be a Čech–Stone compactification of X.Note that if Y is a one-point local compactification of X and FY is a clopen setin βX \X such that Y is equivalent with X ∪ {{FY }}, thenτ(Y ) = {X ∩ U : U ⊃ FY and U open in βX}. (*)Following this notation consider FY1 and FY2 such that Y1 and Y2 are equivalentto X ∪ {{FY1}} and X ∪ {{FY2}} respectively.Property (2) follows easily from (*).To see that {U1 ∪ U2 : U1 ∈ τ(Y1), U2 ∈ τ(Y2)} = τ(Y1) ∩ τ(Y2), take any U1 ∈τ(Y1), U2 ∈ τ(Y2). U2 = (U2 ∪ {∞Y2}) ∩ X is open in X, and thus open in Y1.U1∪{∞Y1} is also open in Y1 and thus so is U1∪{∞Y1}∪U2. Similarly, U1∪{∞Y2}∪U2is open in Y2. The reverse inclusion is trivial.We turn to (1). If FY1 ∩ FY1 = ∅ we have Y1 ∧ Y2 = 0, assume the contrary.Consider U open in βX such that FY1 ∩ FY1 ⊂ U and take V1, V2 open in βX suchBoolean Algebra of One-Point Local Compactifications 117that V1 ∩ V2 = ∅, and we have FY1 \ U ⊂ V1 and FY2 \ U ⊂ V2. Then U1 := V1 ∪ Uand U2 := V2 ∪ U are open (in βX) supersets of respectively FY1 and FY2 such thatU1 ∩ U2 = U , which gives us (1).We are left with (3). To see thatτ(\Y1) ⊂ {X\F : F ⊂ X, for any U ∈ τ(Y1) F\U compact},consider V open in βX such that (βX \X) \ FY1 ⊂ V and take any U open in βXsuch that FY1 ⊂ U . Then (X \ V ) \ U = X \ (U ∪ V ) = βX \ (U ∪ V ) is a closedsubset of βX contained in X and therefore compact.For the reverse inclusion, let V0 and W0 be open sets with disjoint closures inβX such that (βX \ X) \ FY1 ⊂ V0 and FY1 ⊂ W0. Consider F ⊂ X such thatfor any U ∈ τ(Y1) the set F\U is compact. Take any x ∈ X and its closed (in X)neighborhood G such that G is compact. Then X \G ∈ τ(Y1), so F ∩G is compact.Since x and its neighborhood G were arbitrary, this implies that F is closed in X(since if we take x from the boundary of F , we get that it must be in F ). Similarly,since F∩V0 ⊂ F \W0 and W0∩X ∈ τ(Y1), we get that F∩V0 is compact which impliesthat F ∪FY1 is closed in βX. Therefore we have X \F = X∩(βX \(F ∪F0)) ∈ τ(\Y1)which ends the proof of (3).Properties (a) – (c) follow easily from (1) – (3).On the other hand, one can wonder when a family F of sets open in a locally com-pact Hausdorff space X induces its one-point local compactifiaction. The followingproposition answers that question.Proposition 2. Let F be a filter of open sets in a locally compact Hausdorff spaceX. Then F induces a one-point local compactification Y of X such that τ(Y ) = Fiff:1.⋂F = ∅,2. there exists U ∈ F such that for every V ∈ F , U\V is compact,3. for every U ∈ F there exists V ∈ F such that V ⊂ U .Proof. It follows from the definition of τ(Y ) and the definition of a locally compactHausdorff space that those conditions are necessary. We will prove that they are alsosufficient. We take Y := X∪{∞Y }. A set is open in Y iff it is open in X or it is of theform U ∪{∞Y } for some U ∈ F . It follows from (1) and (3) that the topology definedlike that is Hausdorff. It remains to show that Y is locally compact. Take U ∈ Fsuch that for every V ∈ F U\V is compact and assume that U (closure taken in Y ) isnot compact. It follows that there exists a net (xγ) ⊂ U with no convergent subnets.In particular, (xγ) is not convergent to ∞Y , so there exists V1 a neighborhood of∞Y and (yγ) a subnet of (xγ) such that (yγ) ⊂ U\V1 with no convergent subnets, acontradiction.We will now provide a characterization for B(Rn). To this end, we will need factsabout n-point Hausdorff compactifications (see [5] or [3, Theorem 6.8]).118 A. PolańskiTheorem 2 (Theorem 2.1 in [5]). The following statements concerning a space Xare equivalent:1. X has a N -point compactification.2. X is locally compact and contains a compact subset K whose complement is theunion of N mutually disjoint, open subsets {Gi}Ni=1 such that K ∪ Gi is notcompact for each i.3. X is locally compact and contains a compact subset K whose complement isthe union of N mutually disjoint, open subsets {Gi}Ni=1 such that K ∪ Gi iscontained in no compact subset for each i.Using this, we can prove the following facts.Lemma 1. Let X be a locally compact, noncompact Hausdorff space such that forany K ⊂ X compact there exists K0 compact such that K ⊂ K0 and X\K0 hasexactly n connected components (for some fixed n ∈ N independent of the choiceof K), all of them are open and have noncompact (in X) closures. Then X hasan n-point Hausdorff compactification and does not have an (n + 1)-point Hausdorffcompactification.Lemma 2. Let n ∈ N and X be a Hausdorff topological space that has an n-pointHausdorff compactification and does not have an (n+ 1)-point Hausdorff compactifi-cation. Then X is locally compact and |B(X)| = 2n.We will start with Lemma 1.Proof. Applying the assumption of the lemma to the empty set we get that thereexists n ∈ N and K0 compact such that X\K0 has exactly n connected components,let us denote them by G1, . . . , Gn. Therefore (by [5]) X has an n-point Hausdorffcompactification. Suppose that X has an (n+ 1)-Hausdorff compactification. Againby [5], there exist H1, . . . ,Hn+1 such that K1 := X \⋃n+1i=1 Hi is compact, but for eachi the set K1 ∪Hi is not compact. Applying the assumption of the lemma again, thistime to K1, we get that there exists a compact set K2 such that K1 ⊂ K2 and X \K2has n connected components, let us denote them by V1, . . . , Vn. Then there existi0 ∈ {1, . . . , n} and j1, j2 ∈ {1, . . . , n + 1} such that j1 6= j2 and Hi0 has nonemptyintersection with both Vj1 , Vj2 , so it cannot be connected, a contradiction.Now we turn to Lemma 2.Proof. Since X has an n-point Hausdorff compactification, but does not have ann+ 1-point Hausdorff compactification, βX \X has exactly n connected components.From the proof of Theorem 1 we know that |B(X)| = |A(X)|. Each element of A(X)is a union of some connected components of βX \X, so |B(X)| = |A(X)| = 2n.Remark 2. Note that if we assume that if X is a locally compact space such that|B(X)| = 2n, we also get that X has an n-point Hausdorff compactification and doesnot have an (n+ 1)-point Hausdorff compactification (see also [3, Theorem 6.32]).Boolean Algebra of One-Point Local Compactifications 119From the above lemmas we immediately get the following.Corollary 1.• B(R) = {R, [−∞,∞), (−∞,∞],S1}.• B(Rn) = {Rn,Sn} for n > 2.We will now define the end of manifolds, as seen in [2].Definition 4. Let L be a noncompact, connected manifold. Denote by {Kα}α∈K thefamily of all compact subsets of L. We consider descending chainsUα1 ! Uα2 ! · · · ! Uαn ! · · ·where each Uαk is a connected component of L\Kαk , has noncompact closure in L,satisfies Uαk ! Uαk+1 and∞⋂k=1Uαk = ∅.We say that two such chains U = {Uαk}∞k=1 and V = {Uβk}∞k=1 are equivalent (U ∼ V)if for each k > 1 there is n > k such that Uαk ⊃ Vβn and Vβk ⊃ Uαn . It is easy tocheck that ∼ is an equivalence relation. IfU = {Uα1 ! Uα2 ! · · · ! Uαn ! · · · }is as above, we call its equivalence class under ∼ an end of L.Corollary 2.If L is a noncompact, connected, second countable manifold with n ends, n <∞, then|B(L)| = 2n.Proof. LetU1 = {U1α1 ! U1α2 ! · · · }...Un = {Unα1 ! Unα2 ! · · · }be representatives of the ends of L.For every k ∈ {1, 2, . . .}, l ∈ {1, 2, . . . , n} let Klαk be a compact set such that Ulαkis a connected component of L\Klαk . We will show that by taking subsequencesof U2, . . . ,Un we can assume that U l2αk ⊂ L\Kl1αkfor every l2 > l1 (note that asubsequence of a representative of an end is a representative of the same end).Consider K1α1 . Then {L\U2α1 , L\U2α2 , . . .} is an open cover of K1α1 so there existsN1 > 0 such that K1α1 ⊂ L\U2αN1 ⊂ L\U2αN1. Therefore U2αN1 ⊂ L\K1α1 . Similarly,for each m > 1, we can define Nm > Nm−1 such that U2αNm ⊂ L\K1αm . ReplacingU2αm by U2αNmfor each m > 0 we get a subsequence we want for U2. Now we proceedsimilarly for U3, . . . ,Un.120 A. PolańskiWe will now show that by again taking subsequences we can assume that forevery l1 6= l2 we have U l1α1 ∩ Ul2α1 = ∅. Assume the contrary. Then, without loss ofgenerality, for each k > 0 we have U1αk ∩ U2αk6= ∅. Since U2αk ⊂ L\K1αk, the set U2αkis connected, U1αk is a connected component of L\K1αkand U1αk ∩ U2αk6= ∅, it followsthat U2αk ⊂ U1αkfor each k > 0. Now consider K2αk . As before, there exists Nk > ksuch that K2αk ⊂ L\U1αNk. It follows that U1αNk ⊂ L\K2αk. If U1αNk 6⊂ U2αkthenU1αNk∩ U2αk = ∅, so U1αNk∩ U2αNk = ∅ (since U2αNk⊂ U2αk ). Therefore U1αNk⊂ U2αkand so U1 and U2 are representatives of the same end, a contradiction.Now our aim is to use Lemmas 1 and 2, which will end the proof. To this end,we will construct a family of compact sets {Kj}∞j=1. We will need some propertiesof manifolds, namely that a second countable manifold is metrizable and that theone-point compactification of a connected manifold is locally connected (see [4] or [6,page 104]). Let ωL = L ∪ {∞} be the one-point compactification of L. Since L issecond countable we can choose a countable basis of its topology B = {B1, B2, B3, . . .}consisting of open sets with compact closures. Take A1 := K1α1 ∪ . . .∪Knα1 ∪B1. LetK1 be a compact superset of A1 such that ωL\K1 is connected (it exists because ωLis locally connected). Note that connected components of L\K1 are all open and havenoncompact (in L) closures (because ∞ is in the closure taken in ωL of every one ofthem). Again, because L is locally compact we can take an open set A2 with compactclosure such that K1 ∪ B2 ⊂ A2. Let K2 be a compact superset of A2 such thatωL\K2 is connected. As before, all connected components of L\K2 are open and havenoncompact (in L) closures. Moreover, each of them is contained together with itsclosure in some connected component of L\K1. Note that since ωL\K2 has non-emptyintersection with every connected component of L\K1 (because ∞ is in the closuretaken in ωL of every one of them), for every connected component of L\K1 there is atleast one connected component of L\K2 contained in it. Continuing in this manner,we get {Kj}∞j=1. Note that Kj is contained in the interior of Kj+1 for each j > 1and⋃∞j=1Kj = L. Moreover, when j increases the number of connected componentsof L\Kj either increases or stays the same. Consider a connected component U1 ofL\K1. We want to show that U1∩U iα1 6= ∅ for some i. Indeed, otherwise by choosinga connected component U2 of U1 \K2, then a connected U3 of U2 \K3 etc. we wouldget a representative of an end that is not among U1, . . . ,Un, a contradiction. Supposethat U1 ∩ U1α1 6= ∅. Since K1α1 ⊂ K1 and U1α1 , U1 are connected components of theircomplements we get U1 ⊂ U1α1 . The sets Uiα1 are pairwise disjoint, so L \K1 has atleast n connected components. Moreover, the number of connected components ofL\Kj cannot increase past n for any j. Indeed, if we had at least n + 1 connectedcomponents of L \ Kj for some j, we could construct at least n + 1 different ends(similarly as before) which again contradicts the fast that U1, . . . ,Un are all of theends in L. Lemma 1 ends the proof.From this and Remark 2 we also get the following.Corollary 3. If L is a noncompact, connected, second countable manifold with nends, n <∞, then L has an n-point Hausdorff compactification and does not have an(n+ 1)-point Hausdorff compactification.Boolean Algebra of One-Point Local Compactifications 1214. AcknowledgmentsI would like to thank the referees for their valuable remarks, which helped to simplifysome proofs and suggestions that greatly improved the paper.References[1] P. Alexandroff, Über die Metrisation der im Kleinen kompakten topologischenRäume, Math. Ann. 92 (1924) 294–301.[2] A. Candel, L. Conlon, Foliations I, Amer. Math. Soc., 1999.[3] R. Chandler, Hausdorff Compactifications, Marcel Dekker, New York, 1976.[4] J. De Groot, R.H. McDowell, Locally connected spaces and their compactifica-tions, Illinois J. Math. 11 Issue 3 (1967) 353-364.[5] K.D. Magill Jr., N-point compactifications, Am. Math. Mon. 72 (1965) 1075–1081.[6] R.L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloquium Publications42 (1949).DOI: 10.7862/rf.2020.8Artur Polańskiemail: artur.polanski@uj.edu.plORCID: 0000-0002-3302-2551Department of Functional AnalysisInstitute of Mathematics of the Jagiellonian UniversityPOLANDReceived 29.05.2019 Accepted 04.09.2019

Boolean algebra of one-point local compactifications

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Boolean algebra of one-point local compactifications

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Boolean algebra of one-point local compactifications

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