Commentationes Mathematicae Universitatis Carolinae

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1 Commentationes Mathematicae Universitatis Carolinae Alessandro Fedeli On the cardinality of Hausdorff spaces Commentationes Mathematicae Universitatis Carolinae, Vol. 39 (1998), No. 3, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 1998 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Comment.Math.Univ.Carolin. 39,3(1998) On the cardinality of Hausdorff spaces Alessandro Fedeli Abstract. Theaimofthispaperistoshow, usingthereflectionprinciple, threenew cardinal inequalities. These results improve some well-known bounds on the cardinality of Hausdorff spaces. Keywords: cardinal inequality, Hausdorff space Classification: 54A25 Two of the most known inequalities in the theory of cardinal functions are thehajnal-juhàsz sinequality[7]: For X T 2, X 2 c(x)χ(x) andthe Arhangel skii sinequality[5]: For X T 2, X 2 L(X)t(X)ψ(X). Inthispaperwewillusethelanguageofelementarysubmodels(see[4],[10],[1] and[2]) to establish three new cardinal inequalities which generalize the results mentionedabove. Wereferthereaderto[3],[5],[7]fornotationsandterminologynotexplicitlygiven. χ, c, ψ, t, Land π χ denotecharacter,cellularity, pseudocharacter, tightness, Lindelöf degree and π-character respectively. Definitions. (i)let XbeaHausdorffspace. Theclosedpseudocharacterof X,denoted ψ c (X),isthesmallestinfinitecardinal κsuchthatforevery x Xthereisacollection U x ofopenneighbourhoods of xsuchthat {U: U U x }={x}and U x κ([7]). The Hausdorff pseudocharacter of X, denoted Hψ(X), is the smallest infinite cardinal κsuchthatforevery x Xthereisacollection U x ofopenneighbourhoodsof xwith U x κsuchthatif x y,thereexist U U x, V U y with U V = ([6]). Clearly ψ c (X) Hψ(X) χ(x)foreveryhausdorffspace X. (ii)let Xbeatopologicalspace, ac(x)isthesmallestinfinitecardinal κsuch thatthereisasubset Sof Xsuchthat S 2 κ andforeveryopencollection U in XthereisaV [U] κ with U S {V : V V}. Observethat ac(x) c(x)foreveryspace X. Theorem1. If XisaT 2 -spacethen X 2 ac(x)hψ(x). Proof: Let λ=ac(x)hψ(x), κ=2 λ,let τ bethetopologyon Xandlet S beanelementof[x] κ witnessingthat ac(x) λ. Forevery x Xlet B x be acollectionofopenneighbourhoodsof xwith B x λsuchthatif x ythen

3 582 A. Fedeli thereexist U B x, V B y suchthat U V =,andlet f: X P(τ)bethe mapdefinedby f(x)=b x forevery x X. Let A=κ {S, X, τ, κ, f}andtakeaset Msuchthat M A, M =κand whichreflectsenoughformulastocarryoutourargument.tobemoreprecisewe ask that M reflects enough formulas so that the following conditions are satisfied: (i) C Mforevery C [M] κ ; (ii) B x Mforevery x X M; (iii) if B Xand B Mthen B M; (iv) if A Mthen A M; (v) if Bisasubsetof Xsuchthat X M Band B Mthen X= B; (vi) if E Mand E κthen E M. Observethatby(ii)and(vi) B y Mforevery y X M. Claim: X M(andhence X 2 ac(x)hψ(x) ). Suppose notandtake p X \ M. Let B p = {B α } α<λ, clearly {B α : α < λ} = {p}. Nowfor every α < λlet(x M) α = {y X M: B B y forwhich B B α = }. Forevery y (X M) α chooseab y,α B y suchthat B y,α B α =,clearly U α = {B y,α : y (X M) α }covers(x M) α. Since ac(x) λitfollows thatthereisav α [U α ] λ suchthat(x M) α S {V : V V α }.Observe that p / S {V : V V α }(S Mand S κsoby(vi) S M,moreover {V : V Vα } X \B α ).Wehavealso {V: V V α } M(V Mforevery V V α,soby(iii) V M,therefore {V : V V α } Mand {V : V V α } M by(i),henceby(iv) {V : V V α } M,so {V: V V α } Mby(iii)). Set C α = S {V: V V α }forevery α < λandobservethat C α M(recall that S, {V: V V α } M). Now X M {C α : α < λ},since {C α : α < λ} M({C α : α < λ} M,soby(i) {C α : α < λ} M,henceby (iv) {C α : α < λ} M)itfollowsby(v)that {C α : α < λ}=x. Thisis acontradiction(p / {C α : α < λ}). Corollary2([7]). If XisaT 2 -spacethen X 2 c(x)χ(x). Remark3. TheaboveresultofHajnalandJuhàszhasbeenimprovedalsoby Hodel, in factin [6]it isshownthat X 2 c(x)hψ(x) foreveryhausdorff space X.ItisclearthatTheorem1generalizesalsothisresultofHodel. Nowlet XbetheMichaelline,i.e.let Xbe Rtopologizedbyisolatingthe pointsof R \ Qandleavingthepointsof Qwiththeirusualneighbourhoods. Then Xisanormalspacesuchthat X =2 ac(x)hψ(x) <2 c(x)hψ(x). ObservethatinTheorem1Hψ(X)cannotbereplacedby ψ c (X),infactfor everyinfinitecardinal κthereisat 3 -space Xwith X =κand ψ(x)=c(x)= ac(x)=ω(seee.g.[5]). Definition 4. Let X be a topological space, lc(x) is the smallest infinite cardinal κsuchthatthereisaclosedsubset Fof Xsuchthat F 2 κ andforeveryopen collection Uin XthereisaV [U] κ with U F {V : V V}.

4 On the cardinality of Hausdorff spaces 583 Clearly ac(x) lc(x) c(x)foreveryspace X. Theorem5. If XisaHausdorffspacethen X 2 lc(x)πχ(x)ψc(x). Proof: Let λ=lc(x)π χ (X)ψ c (X)andlet κ=2 λ,let τbethetopologyon X andlet Fbeaclosedsubsetof Xwith F κandwitnessingthat lc(x) λ.for every x Xlet B x bealocal π-baseat xsuchthat B x λ,andlet f: X P(τ) bethemapdefinedby f(x)=b x forevery x X. Let A=κ {F, X, τ, κ, f} andtakeaset M Asuchthat M =κandwhichreflectsenoughformulasso that the conditions(i)-(vi) listed in Theorem 1 are satisfied. Claim: X M(andhence X 2 lc(x)πχ(x)ψc(x) ). Supposenotandtake p X \ M. Let {G α : α λ}beafamilyofopenneighbourhoodsof psuch that {G α : α λ}={p}. Set V α = X \ G α and S α = X M V α for every α λ. Nowlet W α = {B:B B y, y S α B V α },since lc(x) λ itfollowsthatthereisav α [W α ] λ suchthat W α F {V: V V α }. Since S α W α (let y S α and Ubeanopenneighbourhoodof y, y / G α so thereisanopenneighbourhood V of ysuchthat V G α =,let B B y such that B U V, B ( W α ) U and y W α )itfollowsthat S α F {V : V V α };moreover {V: V V α } Mand p / F {V: V V α }. Set C α = {V: V V α },since X M F {C α : α < λ}and F {C α : α < λ} Mitfollowsthat F {C α : α < λ}=x,acontradiction. ByTheorem5itfollowsagainthat X 2 c(x)χ(x) forevery T 2 -space X. Moreover we have the following Corollary6([6]). If XisaT 3 -spacethen X 2 c(x)πχ(x)ψ(x). Remark7. Ageneralizationoftheinequalityincorollary6hasalsobeenobtainedbySunin[8]: X 2 c(x)πχ(x)ψc(x) foreveryhausdorffspace X. NotethateventhisresultisacorollaryofTheorem5. Moreoverif X isthe Michaellinethen X =2 lc(x)πχ(x)ψc(x) <2 c(x)πχ(x)ψc(x). Observealsothat the π-charactercannotbeomittedintheorem5(seethecommentattheendof Remark3). NowletusturnourattentiontotheArhangel skii sinequality: For X T 2, X 2 L(X)t(X)ψ(X). Definitions. Let X be a topological space. (i)([8])asubset Aof X with A 2 κ issaidtobe κ-quasi-denseiffor eachopencover U of X thereexistav [U] κ andab [A] κ suchthat ( V) B = X; ql(x)isthesmallestinfinitecardinal κsuchthat X hasa κ-quasi dense subset. (ii) aql(x)isthesmallestinfinitecardinal κsuchthatthereisasubset Sof Xwith S 2 κ suchthatforeveryopencover Uof XthereisaV [U] κ with X= S ( V). Clearly aql(x) L(X)foreveryspace X.

5 584 A. Fedeli Theorem8. If XisaHausdorffspacethen X 2 aql(x)t(x)ψc(x). Proof:Let λ=aql(x)t(x)ψ c (X), κ=2 λ,let τbethetopologyon Xandlet Sbeanelementof[X] κ witnessingthat aql(x) λ.forevery x Xlet B x be afamilyofopenneighbourhoodsof xwith B x λand {B: B B x }={x}, andlet f: X P(τ)bethemapdefinedby f(x)=b x forevery x X. Let A=κ {S, X, τ, κ, f}andtakeaset M Asuchthat M =κandwhich reflects enough formulas so that the conditions(i) (vi) listed in Theorem 1 are satisfied. Firstobservethat X Misaclosedsubsetof X,althoughthisfact followsfromageneralresultwhichcanbefoundin[4]wegiveaproofofitforthe sakeofcompleteness:let x X M,since t(x) λthereisac [X M] λ suchthat x C.Since C M(by(i)),itfollowsthat C M(by(iii)).Nowit remainstoobservethat C κ(recallthat t(x)ψ c (X) λ)andhenceby(vi) x C X M. Wehavedoneifweshowthat X M.Supposethereisap X \M,forevery y X Mlet B y B y suchthat p / B y.since U= {B y : y X M} {X\M}is anopencoverof Xand aql(x) λthereisav [U] λ suchthat X= S ( V). Let W= {B y : B y V},since X M S ( W)and S ( W) Mitfollows that X= S ( W),acontradiction(p / S ( W)). A consequence of Theorem 8 is the following generalization of the Arhangel skii s inequality. Corollary9([8]). If XisaHausdorffspacethen X 2 ql(x)t(x)ψc(x). Proof:Itisenoughtonotethat aql(x) ql(x)t(x)ψ c (X). Remark10. Let κbeaninfinitecardinalnumberandlet Xbethediscretespace ofcardinality2 κ.thisspaceshowsthattheorem8cangiveabetterestimation thantheoneincorollary9. References [1] Dow A., An introduction to applications of elementary submodels to topology, Topology Proc. 13(1988), [2] Dow A., More set-theory for topologists, Topology Appl. 64(1995), [3] Engelking R., General Topology. Revised and completed edition, Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, [4] Fedeli A., Watson S., Elementary submodels and cardinal functions, Topology Proc. 20 (1995), [5] Hodel R.E., Cardinal Functions I, Handbook of Set-theoretic Topology,(Kunen K. and Vaughan J.E., eds.), Elsevier Science Publishers, B.V., North Holland, 1984, pp [6] Hodel R.E., Combinatorial set theory and cardinal functions inequalities, Proc. Amer. Math. Soc. 111(1991), [7] Juhàsz I., Cardinal Functions in Topology ten years later, Mathematical Centre Tracts 123, Amsterdam, [8] Sun S.H., Two new topological cardinal inequalities, Proc. Amer. Math. Soc. 104(1988),

6 On the cardinality of Hausdorff spaces 585 [9] Watson S., The construction of topological spaces: Planks and Resolutions, Recent Progress in General Topology,(Hušek M. and Van Mill J., eds.), Elsevier Science Publishers, B.V., North Holland, 1992, pp [10] Watson S.,The Lindelöfnumber of apower; anintroduction totheuse of elementary submodels in general topology, Topology Appl. 58(1994), Department of Mathematics, University of L Aquila, via Vetoio (Loc. Coppito), L Aquila, Italy fedeli@axscaq.aquila.infn.it (Received September 11, 1997)