582 A. Fedeli there exist U 2B x, V 2B y such that U ë V = ;, and let f : X!Pèçè bethe map dened by fèxè =B x for every x 2 X. Let A = ç ëfs; X; ç; ç;

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1 Comment.Math.Univ.Carolinae 39,3 è1998è581í On the cardinality of Hausdor spaces Alessandro Fedeli Abstract. The aim of this paper is to show, using the reection principle, three new cardinal inequalities. These results improve some well-known bounds on the cardinality of Hausdor spaces. Keywords: cardinal inequality, Hausdor space Classication: 54A25 Two of the most known inequalities in the theory of cardinal functions are the Hajnal-Juhçasz's inequality ë7ë: ëfor X 2 T 2, jxj ç 2 cèxèçèxè " and the Arhangel'skii's inequality ë5ë: ëfor X 2 T 2, jxjç2 LèXètèXèèèXè ". In this paper we will use the language of elementary submodels èsee ë4ë, ë10ë, ë1ë and ë2ëè to establish three new cardinal inequalities which generalize the results mentioned above. We refer the reader to ë3ë, ë5ë, ë7ë for notations and terminology not explicitly given. ç, c, è, t, L and ç ç denote character, cellularity, pseudocharacter, tightness, Lindelíof degree and ç-character respectively. Denitions. èiè Let X be a Hausdor space. The closed pseudocharacter of X, denoted è c èxè, is the smallest innite cardinal ç such that for every x 2 X there is a collection U x of open neighbourhoods of x such that T fu : U 2U x g = fxg and ju x jçç èë7ëè. The Hausdor pseudocharacter of X, denoted HèèXè, is the smallest innite cardinal ç such that for every x 2 X there is a collection U x of open neighbourhoods of x with ju x jçç such that if x 6= y, there exist U 2U x, V 2U y with U ë V = ; èë6ëè. Clearly è c èxè ç HèèXè ç çèxè for every Hausdor space X. èiiè Let X be a topological space, acèxè is the smallest innite cardinal ç such that there is a subset S of X such that jsjç2 ç and for every open collection U in X there is a V2ëUë çç with S UçS ë S fv : V 2Vg. Observe that acèxè ç cèxè for every space X. Theorem 1. If X is a T 2 -space then jxjç2 acèxèhèèxè. Proof: Let ç = acèxèhèèxè, ç =2 ç, let ç be the topology on X and let S be an element ofëxë çç witnessing that acèxè ç ç. For every x 2 X let B x be a collection of open neighbourhoods of x with jb x jçç such that if x 6= y then

2 582 A. Fedeli there exist U 2B x, V 2B y such that U ë V = ;, and let f : X!Pèçè bethe map dened by fèxè =B x for every x 2 X. Let A = ç ëfs; X; ç; ç; fg and take a set M such that MçA, jmj = ç and which reects enough formulas to carry out our argument. To be more precise we ask that M reects enough formulas so that the following conditions are satised: èiè C 2Mfor every C 2 ëmë çç ; èiiè B x 2Mfor every x 2 X ëm; èiiiè if B ç X and B 2Mthen B 2M; èivè if A2Mthen S A2M; èvè if B is a subset of X such that X ëmçb and B 2Mthen X = B; èviè if E 2Mand jejçç then E çm. Observe that by èiiè and èviè B y çmfor every y 2 X ëm. Claim: X ç M èand hence jxj ç 2 acèxèhèèxè è. Suppose not and take p 2 X nm. Let B p = fb g T éç, clearly fb : é çg = fpg. Now for every éçlet èx ëmè = fy 2 X ëm: 9B 2B y for which B ë B = ;g. For every y 2 èx ëmè choose a B y; 2B y such that B y; ë B = ;, clearly U = fb y; : y 2 èx ëmè g covers èx ëmè. Since acèxè ç ç it follows that there is a V 2 ëu ë çç such that èx ë Mè ç S ë S fv : V 2V g. Observe that p =2 S ë S fv : V 2V g ès 2Mand jsj çç so by èviè S çm, moreover S fv : V 2V gçxn S B è. We have also fv : V 2V g2mèv 2Mfor every V 2V,soby èiiiè V 2M, S therefore fv : V 2V gçmand fv : V 2V g2m by èiè, hence by èivè fv : V 2 V g 2 S M, so fv : V 2V g 2 M by èiiièè. Set C = S ë S S fv : V 2V g for every éçand observe that C 2Mèrecall that S; fv : V 2V g 2 Mè. Now X ëmç S S fc : é çg, since fc : é S çg 2 M èfc : é çg ç M, so by èiè fc S : é çg 2 M, hence by èivè fc : éçg2mèitfollows by èvè that fc : éçg = X. This is a contradiction èp =2 S fc : éçgè. Corollary 2 èë7ëè. If X is a T 2 -space then jxjç2 cèxèçèxè. Remark 3. The above result of Hajnal and Juhçasz has been improved also by Hodel, in fact in ë6ë it is shown that jxj ç 2 cèxèhèèxè for every Hausdor space X. It is clear that Theorem 1 generalizes also this result of Hodel. Now let X be the Michael line, i.e. let X be R topologized by isolating the points of R n Q and leaving the points of Q with their usual neighbourhoods. Then X is a normal space such that jxj =2 acèxèhèèxè é 2 cèxèhèèxè. Observe that in Theorem 1 HèèXè cannot be replaced by è c èxè, in fact for every innite cardinal ç there is a T 3 -space X with jxj = ç and èèxè =cèxè = acèxè =! èsee e.g. ë5ëè. Denition 4. Let X be a topological space, lcèxè is the smallest innite cardinal ç such that there is a closed subset F of X such that jfjç2 ç and for every open collection U in X there is a V2ëUë çç with S UçF ë S fv : V 2Vg.

3 On the cardinality of Hausdor spaces 583 Clearly acèxè ç lcèxè ç cèxè for every space X. Theorem 5. If X is a Hausdor space then jxjç2 lcèxèççèxèècèxè. Proof: Let ç = lcèxèç ç èxèè c èxè and let ç =2 ç, let ç be the topology on X and let F be a closed subset of X with jfjçç and witnessing that lcèxè ç ç. For every x 2 X let B x be a local ç-base at x such that jb x jçç, and let f : X!Pèçè be the map dened by fèxè =B x for every x 2 X. Let A = ç ëff; X; ç; ç; fg and take a set MçA such that jmj = ç and which reects enough formulas so that the conditions èiè-èviè listed in Theorem 1 are satised. Claim: X çmèand hence jxj ç2 lcèxèççèxèècèxè è. Suppose not and take p 2 X nm. Let fg : 2 çg be a family of open neighbourhoods of p such that T fg : 2 çg = fpg. Set V = X n G and S = X ëmëv for every 2 ç. Now let W = fb : B 2B y ;y 2 S ^ B ç V g, since lcèxè ç ç it follows that there is a V 2 ëw S ë çç such that W ç F ë S fv : V 2V g. Since S ç S W èlet y 2 S and U be an open neighbourhood of y, y =2 G so there is an open neighbourhood V of y such that V ë G = ;, let B 2B y such that B ç U ë V, ; 6= B ç S è W è ë U and y 2 S W è it follows that S ç F ë S fv : V 2V S g; moreover fv : V 2V g2mand p=2 F ë S S fv : V 2V g. Set C = fv : V 2V g, since X ëmçf ë S fc : éçg and F ë S fc : éçg2mit follows that F ë S fc : éçg = X, a contradiction. By Theorem 5 it follows again that jxj ç 2 cèxèçèxè for every T 2 -space X. Moreover we have the following Corollary 6 èë6ëè. If X is a T 3 -space then jxjç2 cèxèççèxèèèxè. Remark 7. A generalization of the inequality incorollary 6 has also been obtained by Sun in ë8ë: ëjxj ç 2 cèxèççèxèècèxè for every Hausdor space X". Note that even this result is a corollary of Theorem 5. Moreover if X is the Michael line then jxj =2 lcèxèççèxèècèxè é 2 cèxèççèxèècèxè. Observe also that the ç-character cannot be omitted in Theorem 5 èsee the comment at the end of Remark 3è. Now let us turn our attention to the Arhangel'skii's inequality: ëfor X 2 T 2, jxjç2 LèXètèXèèèXè ". Denitions. Let X be a topological space. èiè èë8ëè A subset A of X with jaj ç 2 ç is said to be ç-quasi-dense if for each open cover U of X there exist a V 2 ëuë çç and a B 2 ëaë çç such that è S Vè ë B = X; qlèxè is the smallest innite cardinal ç such that X has a ç-quasi dense subset. èiiè aqlèxè is the smallest innite cardinal ç such that there is a subset S of X with jsjç2 ç such that for every open cover U of X there is a V2ëUë çç with X = S ë è S Vè. Clearly aqlèxè ç LèXè for every space X.

4 584 A. Fedeli Theorem 8. If X is a Hausdor space then jxjç2 aqlèxètèxèècèxè. Proof: Let ç = aqlèxètèxèè c èxè, ç =2 ç, let ç be the topology on X and let S be an element ofëxë çç witnessing that aqlèxè ç ç. For every x 2 X let B x be a family of open neighbourhoods of x with jb T x jçç and fb : B 2B x g = fxg, and let f : X!Pèçè be the map dened by fèxè =B x for every x 2 X. Let A = ç ëfs; X; ç; ç; fg and take a set M ç A such that jmj = ç and which reects enough formulas so that the conditions èièíèviè listed in Theorem 1 are satised. First observe that X ëm is a closed subset of X, although this fact follows from a general result which can be found in ë4ë we give a proof of it for the sake of completeness: let x 2 X ëm, since tèxè ç ç there is a C 2 ëx ëmë çç such that x 2 C. Since C 2Mèby èièè, it follows that C 2Mèby èiiièè. Now it remains to observe that jcj çç èrecall that tèxèè c èxè ç çè and hence by èviè x 2 C ç X ëm. We have done if we show that X çm. Suppose there is a p 2 X nm, for every y 2 XëM let B y 2B y such that p=2 B y. Since U = fb y : y 2 XëMgëfXnMg is an open cover of X and aqlèxè ç S ç there is a V2ëUë çç such that X = Sëè Vè. Let W = fb y : B y 2Vg, since X ëm ç S S S ëè Wè and S ëè Wè 2Mit follows that X = S ë S è Wè, a contradiction èp =2 S ë S è Wèè. A consequence of Theorem 8 is the following generalization of the Arhangel'skii's inequality. Corollary 9 èë8ëè. If X is a Hausdor space then jxjç2 qlèxètèxèècèxè. Proof: It is enough to note that aqlèxè ç qlèxètèxèè c èxè. Remark 10. Let ç be an innite cardinal number and let X be the discrete space of cardinality 2 ç. This space shows that Theorem 8 can give a better estimation than the one in Corollary 9. References ë1ë Dow A., An introduction to applications of elementary submodels to topology, Topology Proc. 13 è1988è, 17í72. ë2ë Dow A., More set-theory for topologists, Topology Appl. 64 è1995è, 243í300. ë3ë Engelking R., General Topology. Revised and completed edition, Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, ë4ë Fedeli A., Watson S., Elementary submodels and cardinal functions, Topology Proc. 20 è1995è, 91í110. ë5ë Hodel R.E., Cardinal Functions I, Handbook of Set-theoretic Topology, èkunen K. and Vaughan J.E., eds.è, Elsevier Science Publishers, B.V., North Holland, 1984, pp. 1í61. ë6ë Hodel R.E., Combinatorial set theory and cardinal functions inequalities, Proc. Amer. Math. Soc. 111 è1991è, 567í575. ë7ë Juhçasz I., Cardinal Functions in Topology í ten years later, Mathematical Centre Tracts 123, Amsterdam, ë8ë Sun S.H., Two new topological cardinal inequalities, Proc. Amer. Math. Soc. 104 è1988è, 313í316.

5 On the cardinality of Hausdor spaces 585 ë9ë Watson S., The construction of topological spaces: Planks and Resolutions, Recent Progress in General Topology, èhuçsek M. and Van Mill J., eds.è, Elsevier Science Publishers, B.V., North Holland, 1992, pp. 675í757. ë10ë Watson S., The Lindelíof number of a power; an introduction to the use of elementary submodels in general topology, Topology Appl. 58 è1994è, 25í34. Department of Mathematics, University of L'Aquila, via Vetoio èloc. Coppitoè, L'Aquila, Italy fedeli@axscaq.aquila.infn.it èreceived September 11, 1997è