Measurable cardinals and the cardinality of Lindel"of spaces

MEASURABLE CARDINALS AND THE CARDINALITY OF LINDEL ¨ OF SP A CES MARION SCHEEPERS Abstract. W e obtain fr om the consistency of the existence of a measurable cardinal the consistency of “smal l” upp er b ounds on the cardinality of a large class of Lindel¨ of spaces whose s ingletons are G δ sets. Call a top ological space in whic h e a ch singleton is a G δ set a p oints G δ sp ac e . A.V. Arhangel’skii prov ed that any p oints G δ Lindel¨ of space m ust hav e car dinality less than the least measura ble car dina l and asked whether for T 2 spaces this cardinalit y upper b ound could be impro ved. I. Juhasz constructed examples showing that for T 1 spaces this upp er bound is sharp. F.D. T all, inv estigating Arhangel’sk ii’s problem, defined the class of indestruct ibly Lindel¨ of spaces . A L indel¨ o f space is indestructible if it remains Lindel¨ of after forcing with a countably closed forcing notion. He prov ed: Theorem 1 (F.D. T all [16]) . If it is c onsistent that ther e is a sup er c omp act c ar dinal, then it i s c onsistent t hat 2 ℵ 0 = ℵ 1 , and every p oints G δ indestructibly Lindel¨ of sp ac e has c ar dinality ≤ ℵ 1 . In this pape r w e show that the h yp othesis that ther e is a supe r compact cardinal can be weakened to the h yp othesis that there exists a measurable cardinal. Our techn ique p ermits flexibility on the cardinality o f the contin uum. In Section 1 we review relev ant information ab out ideals and the weakly pre- cipitous idea l ga me. The re lev ance of the weakly precipitous ideal g ame to p oints G δ spaces is g iven in Lemma 2. In Section 2 we cons ide r the indestructibly Lin- del¨ of spaces. A v a riation of the weakly precipitous ideal game is in tro duced. This v aria tio n is featured in the main result, Theorem 4: a cardinality restrictio n is im- po sed o n the indestructibly Lindel¨ of spaces with points G δ . In Section 3 w e give the consistency str ength of the hypo thesis used in Theorem 4 a nd point out that mere existence of a precipitous ideal is insufficient to deriv e a cardinalit y b ound on the indestructibly Lindel¨ of p oints G δ spaces. In Section 4 w e describ e mo dels of set theory in whic h the Con tinuum Hyp othesis fails while ther e is a “ small” upper bo und on the cardinality of points G δ indestructibly Lindel¨ of spaces. The notio n of a Ro th b erger space app ears in the pa per . A space X is a R othb er ger sp ac e if for e ach ω - sequence of open covers of X there is a seq uence of op en sets, then n -th b elong ing to the n -th cov er, such that the terms of the latter s e quence is an op en cov er o f X . Ro th b erger spaces are indestructibly Lindel¨ of (but no t conv ersely). More details ab out Rothberger spaces in this co n text can b e found in [14]. Date : V ersi on of January 29, 2010. 1 2 MARION SCHEEP ERS Ackno wledgement I would like to thank Pr ofessor F.D. T a ll for v ery a s tim ulating co rresp ondence regar ding the problems trea ted in this pap er, a nd for permissio n to include his argument (slig ht ly adapted) in the pro of o f Theore m 13. I w ould also lik e to thank Professo r Masaru Kada for imp o rtant remarks ab out Theorem 4, and for permissio n to include his remar ks (See remark (3) in the last section of the pap er). 1. W eakl y precipitous ideals and points G δ sp aces. F or κ an infinite cardina l, P ( κ ) denotes the p ow er set of κ . A set J ⊆ P ( κ ) is said to b e a fr e e ide al on κ if: (i) each finite subset of κ is an ele ment of J , (ii) κ is not a n elemen t of J , (iii) the union of any tw o elements of J is an e lemen t of J , a nd (iv) if B ∈ J then P ( B ) ⊂ J . F or a free ideal J the s y m b ol J + denotes { A ∈ P ( κ ) : A 6∈ J } . Let λ ≤ κ be a cardinal num b er. The free idea l J on κ is s aid to b e λ - c omplete if: F or ea ch A ⊂ J , if |A| < λ then S A ∈ J . A free ideal whic h is ω 1 -complete is said to be σ -c o mplete. F or a free ideal J on κ Ga lvin et al. [2] inv es tigated the ga me G ( J ) of length ω , defined as follows: Two pla yers, ONE and TWO, play an inning per finite ordinal n . In inning n ONE first chooses O n ∈ J + . TW O r esp onds with T n ∈ J + . The play e r s o bey the rule that for each n , O n +1 ⊂ T n ⊂ O n . TWO wins a pla y O 1 , T 1 , O 2 , T 2 , · · · , O n , T n , · · · if T n<ω T n 6 = ∅ ; else, ONE wins. It is easy to see that if J is not σ -complete, then ONE has a winning strategy in G ( J ). It was s hown in Theorem 2 of [2] that J ⊆ P ( κ ) is a we akly pr e cipitous ideal if, and o nly if, ONE has no winning strategy in the g ame G ( J ). W e shall take this characterization of weak precipitous ness as the definition. An ideal J on P ( κ ) is said to be pr e cipitous if it is weakly pr ecipitous and κ -complete. This distinction w a s no t made in the earlier litera ture such as [2] and [7]. The κ -completeness require ment app ears to hav e emer ged in [8], and the “weakly precipito us” terminology for the σ -complete case seems to have b een coined in [11]. This game is related as follows to spaces in whic h each point is G δ : Lemma 2. L et κ b e a c ar dinal s uch that ther e is a we akly pr e cipitous ide al J ⊂ P ( κ ) . L et X ⊇ κ b e a top olo gic al sp ac e in wh ich e ach p oint is a G δ . Then for e ach x ∈ X and e ach B ∈ J + and e ach se quenc e ( U n ( x ) : n < ω ) of neighb orho o ds of x with { x } = ∩ n<ω U n ( x ) , ther e is a C ⊆ B with C ∈ J + and an n su ch that U n ( x ) ∩ C ∈ J Pro of: F or each x in X fix a seq uenc e ( U n ( x ) : n < ω ) of op en neighborho o ds such that fo r e a ch n we ha ve U n +1 ( x ) ⊆ U n ( x ), and { x } = ∩ n<ω U n ( x ). Suppo se that contrary to the claim of the lemma, there is an x ∈ X and a B ∈ J + such that for e a ch C ⊂ B with C ∈ J + and for ea ch n, U n ( x ) ∩ C ∈ J + . Fix x and B . Define a stra tegy σ for ONE of the ga me G 1 ( J ) a s follows: ONE’s first mov e is σ ( X ) = ( U 1 ( x ) ∩ B ) \ { x } . When TWO resp onds with a T 1 ⊆ σ ( X ) and T 1 ∈ J + , ONE pla ys σ ( T 1 ) = ( U 2 ( x ) ∩ T 1 ) \ { x } . When TWO res po nds with T 2 ⊆ σ ( T 1 ), ONE plays σ ( T 1 , T 2 ) = ( T 2 ∩ U 3 ( x )) \ { x } , and s o o n. MEASURABLE CARDINALS AND THE CARDINALITY OF LINDEL ¨ OF SP ACES 3 Observe that σ is a legitimate strategy for ONE. But since ONE has no winning strategy in G ( J ), consider a σ -play lost b y ONE, say O 1 , T 1 , O 2 , T 2 , · · · Then ∩ ∞ n =1 T n 6 = ∅ , implying that ∩ ∞ n =1 ( U n ( x ) \ { x } ) 6 = ∅ , a contradiction.  2. The card inality of points G δ indestructi bl y L indel ¨ of sp aces. F or a spac e X define the game G ω 1 1 ( O , O ) as fo llows: Players O NE and TWO play an inning for each γ < ω 1 . In inning γ ONE first c ho oses a n o p en co ver O γ of X , and then TWO c ho oses T γ ∈ O γ . A play O 0 , T 0 , · · · , O γ , T γ , · · · is w on by TW O if { T γ : γ < ω 1 } is a cov er of X . Else, ONE wins. In [14] w e prov e d the follo wing characterizatio n of b eing indestructibly Lindel¨ of: Theorem 3 ([1 4], Theorem 1) . A Lindel¨ of sp ac e X is indestructibly Lindel¨ of if, and only if, ONE has no winning str ate gy in the game G ω 1 1 ( O , O ) . Of sev eral natural v a riations o n G ( J ) w e now need the following one: The g a me G + ( J ) pr o ceeds like G ( J ), but TW O wins a play o nly when T n<ω T n ∈ J + ; else, ONE wins. Evidently , if TWO has a winning str ategy in G + ( J ) then TWO ha s a winning strategy in G ( J ). Similarly , if O NE ha s no winning s trategy in G + ( J ), then ONE ha s no winning s tr ategy in G ( J ). A winning str ategy in G + ( J ) for TWO which dep ends on only the most r ecent mov e of ONE is s aid to be a winning t actic . Theorem 4. Assume ther e is a fr e e, σ -c omplete ide al J on κ such t hat TWO has a winning t actic in G + ( J ) . Then e ach p oints G δ indestructibly Lindel¨ of sp ac e has c ar dinality less than κ . Pro of: Let X be a points G δ Lindel¨ of space with | X | ≥ κ . Let Y be s ubset of X of cardinality κ a nd let J ⊂ P ( Y ) b e a fr ee ideal such that TWO ha s a winning ta ctic σ in G + ( J ). W e define a winning stra tegy F for ONE of the game G ω 1 1 ( O , O ) and then cite Theorem 1 of [14]: F or each x ∈ X fix a sequence o f neighbo rho o ds ( U n ( x ) : n < ∞ ) suc h that for m < n we hav e U m ( x ) ⊃ U n ( x ), and { x } = T n<ω U n ( x ). First, ONE do es the following: F or e a ch x ∈ X : Cho ose D x ⊆ Y and n so that D x 6∈ J , U n ( x ) ∩ D x ∈ J and set C ( x ) = σ ( D x ). Cho ose n ( C ( x ) , x ) < ω such that C ( x ) ∩ U n ( C ( x ) ,x ) ∈ J . ONE’s first mov e in G ω 1 1 ( O , O ) is F ( ∅ ) = { U n ( C ( x ) ,x ) ( x ) : x ∈ X } . When TW O ch o ose s T 0 ∈ F ( ∅ ), fix x 0 with T 0 = U n ( C ( x 0 ) ,x 0 ) ( x 0 ). Define C 0 = C ( x 0 ), D 0 = D x 0 . Since C 0 ∈ J + we c ho ose by Lemma 2 for each x ∈ X a D x 0 ,x and an n with: (1) D x 0 ,x ∈ J + and D x 0 ,x ⊂ C ( x 0 ) and (2) U n ( x ) ∩ D x 0 ,x ∈ J . Put C ( x 0 , x ) = σ ( D x 0 ,x ), and c ho ose n ( C ( x 0 , x ) , x ) with C ( x 0 , x ) ∩ U n ( C ( x 0 ,x ) ,x ) ( x ) ∈ J . O NE plays F ( T 0 ) = { U n ( C ( x 0 ,x ) ,x ) ( x ) : x ∈ X } . 4 MARION SCHEEP ERS When TWO plays T 1 ∈ F ( T 0 ), fix x 1 so that T 1 = U n ( C ( x 0 ,x 1 ) ,x 1 ) ( x 1 ). Define C 1 = C ( x 0 , x 1 ) and D 1 = D x 0 ,x 1 . Since C 1 ∈ J + we c ho ose b y Le mma 2 for each x ∈ X a D x 0 ,x 1 ,x and an n with: (1) D x 0 ,x 1 ,x ∈ J + and D x 0 ,x 1 ,x ⊂ C ( x 0 , x 1 ) and (2) U n ( x ) ∩ D x 0 ,x 1 ,x ∈ J . Put C ( x 0 , x 1 , x ) = σ ( D x 0 ,x 1 ,x ) and c ho ose n ( C ( x 0 , x 1 , x ) , x ) with C ( x 0 , x 1 , x ) ∩ U n ( C ( x 0 ,x 1 ,x ) ,x ) ( x ) ∈ J . ONE plays F ( T 0 , T 1 ) = { U n ( C ( x 0 ,x 1 ,x ) ,x ) ( x ) : x ∈ X } , and so on. A t a limit stag e α < ω 1 we have descending sequences ( C γ : γ < α ) a nd ( D γ : γ < α ) of e le ments of J + as well as a sequence ( x γ : γ < α ) o f elemen ts of X such that: (1) F or each γ , C γ = C ( x ν : ν ≤ γ ) and D γ = D ( x ν : ν ≤ γ ) ; (2) F or each γ , D γ +1 ⊂ C γ = σ ( D γ ) (3) T γ = U n ( C γ ,x γ ) ( x γ ). Since α is co untable choos e a cofinal s ubset ( γ n : n < ω ) of o rdinals incr easing to α . Then as for each n w e hav e C γ n = σ ( D γ n ) we see thst ( C γ n : n < ω ) are mov es by TW O, using the winning tactic σ , in G + ( J ). Thus we ha ve ∩ γ <α C γ = ∩ n<ω C γ n ∈ J + . Then by Le mma 2 c ho ose for each x ∈ X a D ( x ν : ν ≤ γ ) ⌢ ( x ) and n such that (1) D ( x ν : ν ≤ γ ) ⌢ ( x ) ∈ J + and D ( x ν : ν ≤ γ ) ⌢ ( x ) ⊂ ∩ γ <α C γ and (2) D ( x ν : ν ≤ γ ) ⌢ ( x ) ∩ U n ( x ) ∈ J . Put C (( x ν : ν < α ) ⌢ ( x )) = σ ( D (( x ν : ν <α ) ⌢ ( x ) ) ) and choos e n ( C (( x ν : ν < α ) ⌢ x ) , x ) s uch that: C (( x ν : ν < α ) ⌢ x ) ∩ U n ( C (( x ν : ν <α ) ⌢x ) ( x ) ∈ J . Then ONE plays F ( T γ : γ < α ) = { U n ( C (( x ν : ν <α ) ⌢x ) ,x ) ( x ) : x ∈ X } . This defines a strategy for ONE of the ga me G ω 1 1 ( O , O ). T o see that it is winning, suppo se that on the contrary there is an F - play w on by TW O , say O 0 , T 0 , · · · , O γ , T γ , · · · , γ < ω 1 , where O 0 = F ( ∅ ) and for each γ > 0, O γ = F ( T β : β < γ ). Since TWO wins G ω 1 1 ( O , O ), X = ∪ γ <ω 1 T γ . But X is Lindel¨ of, and so w e find a β < ω 1 with X = ∪ γ <β T γ . But then C α = C ( x ν : ν < α ), α < β o ccurring in the definition of F are in J + and sa tisfy for α < β that C β ⊂ C α . It follows that fo r each γ < β we have T γ ∩ C β ∈ J , and as J is σ -c omplete if follows that the T γ do not c ov e r C β ⊂ X , a con tradictio n.  A t one point in the a bove proof we made use of the fact that TWO has a winning ta ctic in G + ( J ). It may be the case that the co nclusion of the theorem can be deduced fr o m simply assuming that TWO has a winning strateg y in G + ( J ) 1 . One wa y to a chiev e this would be to show that if TW O ha s a winning stra teg y in G + ( J ), then TW O has a winning tactic in G + ( J ). I hav e not in vestigated this. 1 In fact, the c onclusion of th e theo rem can b e deduced from this for m ally weak er h yp othesis. See Remark (3) at the end of the paper. MEASURABLE CARDINALS AND THE CARDINALITY OF LINDEL ¨ OF SP ACES 5 Problem 1. L et J b e a σ -c omplete fr e e ide al on κ su ch t hat TWO has a winning str ate gy in G + ( J ) . Do es it fol low that TWO has a winning tactic in G + ( J ) ? Note that by Theor em 7 of [3], if TW O has a winning strategy in G + ( J ), then TW O has a winning s trategy σ such that T 1 = σ ( O 1 ), and for each n , T n +1 = σ ( T n , O n +1 ). Here is another na tur al question which I have no t explored: Problem 2. L et J b e a σ -c omplete fr e e ide al on κ su ch t hat TWO has a winning str ate gy in G ( J ) . Do es it fol low that TWO has a winning str ate gy in G + ( J ) ? 3. The hypo thesis “TWO has a winning t actic in G + ( J ) ”. W e no w co nsider the strength o f the h yp othesis that TWO has a winning tactic in G + ( J ). First rec a ll s o me co ncepts. An ideal J is said to b e λ - satur ate d if whe never B ⊂ P ( κ ) \ J is suc h that whenever X 6 = Y ar e ele ments of B then X ∩ Y ∈ J , then |B | < λ . W e write sat ( J ) = min { λ : J is λ -sa tur ated } . When sa t ( J ) is infinite it is regular a nd uncountable. No te that if λ < µ and if J is λ -satura ted then it is µ -satura ted. It is well-known that every κ + -saturated κ -co mplete ideal on κ is precipitous (see Lemma 22.22 of [6]). Next, let J ⊂ P ( κ ) b e a σ -complete ideal and let λ ≤ κ b e a n initial ordinal. F or subsets X and Y of κ write X ≡ Y mo d J if the symmetric difference of X a nd Y is in J . Then P ( κ ) /J denotes the set of equiv ale nce classes for this relation, and [ X ] J denotes the equiv alence clas s of X . A subset D of the Bo olea n algebr a P ( κ ) /J is said to b e dense if there is for each b ∈ P ( κ ) /J a d ∈ D with d < b . A dens e set D ⊂ P ( κ ) /J is said to be λ - dense if for each β < λ , for each β -sequence b 0 > b 1 > · · · > b γ > · · · , γ < β < λ of elements of D there is a d ∈ D such that for a ll γ < β , d < b γ . The Dense Ide al Hyp othesis for κ ≥ λ , denoted DIH ( κ, λ ), is the statement: There is a σ - complete free ideal J ⊂ P ( κ ) such that the Bo olean algebra P ( κ ) /J has a λ -dense subset D . Note that if µ < λ then DIH ( κ, λ ) ⇒ DIH ( κ, µ ). Consider the following fiv e statemen ts: I There exists a measur able cardinal II Ther e is an ω 1 dense free ideal J on an infinite set S . II I There is a fr ee idea l J on a set S such that TW O has a winning tactic in G + ( J ). IV There is a precipitous 2 ideal J on an infinite set S . V There is an κ + -saturated κ -co mplete free ideal on a regular cardinal κ . Then I ⇒ I I (let J be the dual idea l of the ultrafilter witnessing measur ability), II ⇒ II I (see remarks (1), (3) and (4) on pa ge 292 of [2]), and evidently II I ⇒ IV. As already noted, V ⇒ IV. In ZFC, for a statement P , let CON(P) denote “P is consisten t”. Then we have CON(I) if, and only if, CON(IV), and CON(V) implies CO N(I): Prop ositio n 5. The existen c e of a fr e e ide al J on an unc ountable c ar dinal such that TWO has a winn ing tactic in G + ( J ) is e quic onsisten t with the existenc e of a me asur able c ar dinal. 2 Indeed, we akly precipitous works here. 6 MARION SCHEEP ERS Pro of: When TWO has a winning tactic in G + ( J ), then TW O has a winning strategy in G ( J ), a nd thus ONE has no winning s trategy in G ( J ). It follows that J is a weakly precipitous ideal. Jech et al. [7] show that the existence of a weakly precipitous idea l is equiconsistent with the existence of a measur able cardinal. This shows that consistency of the existence of a free ideal J on an uncountable set, such that TWO has a winning tactic in G + ( J ) implies the consistency o f the existence of a measurable cardina l. F or the o ther directio n: The a rgument in § 4 of [2] can b e adapted to show that if it is consisten t that there is a measurable cardinal κ , then for an y infinite cardinal λ < κ it is consistent that DIH ( λ ++ , λ + ) ho lds 3 . A free ideal J on λ ++ witnessing DIH ( λ ++ , λ + ) is a free σ -complete ideal such that TW O has a winning tactic in G + ( J ).  Prop erty V is preserved when adding ℵ 1 or mor e Cohen r eals. This follows from the following well-known fact sta ted as Lemma 22.32 in [6]: Lemma 6. L et B b e a c omplete Bo ole an algebr a, let G b e V -generic on B and let κ b e a r e gular unc ountable c ar dinal. Assume t hat s at ( B ) ≤ λ and sat ( B ) < κ . T hen: If J ⊆ P ( κ ) is λ -satur ate d and κ - c omplete, then in V [ G ] J gener ates a λ -satur ate d κ -c omplete ide al on κ . Now I. Juhas z has prov en that for each infinite cardinal κ less than the fir st measurable car dinal there is a points G δ Lindel¨ of spa c e X with κ < | X | (see [16] for details). But adding ℵ 1 Cohen r eals conv erts each such groundmo del spa ce to a Rothberger space (and thus indestructibly Lindel¨ of space) in the gener ic extension (see [14]). Thus if it is consistent that ther e is a µ + -saturated µ -complete ideal on some reg ula r cardinal µ , then it is co nsistent that there is a (weakly) precipitous ideal on µ , and yet there is a n indestr uc tibly Lindel¨ of p oints G δ space of c ardinality larger than µ . 4. The con tinuum and the cardinality o f points G δ indestructi bl y Lindel ¨ of sp aces. The fir st co nsequence of the work ab ov e is that the hypothesis of the consistency of the exis tence of a sup erc o mpact cardinal in Theo rem 1 can b e reduce d to the hypothesis of the consistency of the ex istence of a measurable cardinal: Corollary 7. If it is c onsistent that ther e is a me asur able c ar dinal, then it is c onsistent that 2 ℵ 0 = ℵ 1 and al l ind estruct ible p oints G δ Lindel¨ of sp ac es ar e of c ar dinality ≤ ℵ 1 . In what follows we demonstrate that the a b ound on the c a rdinality of p oints G δ indestructibly Lindel¨ of spaces do es not have a s trong influence o n the cardinality of the real line. Corollary 8. If it is c onsisten t that ther e is a me asur able c ar dinal κ , then for e ach r e gular c ar dinal ℵ α with κ > ℵ α > ℵ 0 it is c onsistent that 2 ℵ 0 = ℵ α +1 and for al l µ ≤ 2 ℵ 0 ther e ar e indestructible p oints G δ Lindel¨ of sp ac es and ther e ar e none of c ar dinality > 2 ℵ 0 . 3 The model in [ 2] is obtained as f ollows: F or κ a measurable in the ground mo del, coll apse all cardinals b elo w κ to ℵ 1 using the Levy collapse. One can s ho w that with µ < κ an uncountab le regular cardinal, collapsing all cardinals below κ to µ produces a mo del of DIH ( µ + , µ ), by v er ifying that Lemmas 1, 2 and 3 and the subsequen t claims in [2] apply mutatis mutandis . MEASURABLE CARDINALS AND THE CARDINALITY OF LINDEL ¨ OF SP ACES 7 Pro of: First raise the co ntin uum to ℵ α +1 by adding reals. Next L ´ evy co llapse the measurable car dinal to ℵ α +2 . In the r esulting mo del 2 ℵ 0 = ℵ α +1 and DIH ( ℵ α +2 , ℵ α +1 ) holds. By Theor em 1 each indestructibly Lindel¨ of space with p oints G δ has car- dinality ≤ ℵ α +1 in this g eneric extension. Since each separa ble metric space is indestructibly Lindel¨ of it follows that there is for each ca rdinal λ ≤ ℵ α +1 an inde- structibly Lindel¨ of p oints G δ space of cardinality λ .  Corollary 9. If it is c onsisten t that ther e is a me asur able c ar dinal κ , then for e ach p air of re gular c ar dinals ℵ α < ℵ β < κ with ℵ ℵ 1 β = ℵ β it is c onsistent that 2 ℵ 0 = ℵ α and 2 ℵ 1 = ℵ β and t her e is a p oints G δ indestructibly Lindel¨ of sp ac e of c ar dinality ℵ β , bu t t her e ar e no p oints G δ indestructible Lindel¨ of sp ac es of c ar dinality > 2 ℵ 1 . Pro of: W e ma y ass ume the ground mo del is L [ U ] wher e U is a normal ultrafilter witnessing measurability , and thus that GCH holds. First use Gorelic’s cardinal- and cofinality- preserv ing for cing to raise 2 ℵ 1 to ℵ β while maintaining CH. This gives a points G δ indestructibly L indel¨ o f T 3 -space X of cardinality 2 ℵ 1 . Then add ℵ α Cohen r eals to get 2 ℵ 0 = ℵ α . In this extension the space X from the first step still is a p oints G δ indestructibly Lindel¨ of T 3 -space since all these prop erties are preserved by Cohen re a ls [1 4]. The cardinal κ is, in this gener ic extensio n, still measurable [13]. Finally , Levy co llapse the measurable c ardinal to ℵ β +1 . This forcing is countably closed (and more) a nd thus preserves indestructibly Lindel¨ o f spaces from the ground model. The resulting mo del is the one for the corollary .  5. Regarding a problem of Hajnal and Juhasz. Ha jnal and Juhasz asked if an uncountable T 2 -Lindel¨ of space m ust contain a Lindel¨ of subspace o f cardinality ℵ 1 . Baumgartner and T all s how ed in [1] that there are ZFC examples of uncount able T 1 Lindel¨ of spa c es with points G δ which have no Lindel¨ of subspaces of car dina lit y ℵ 1 . In [9 ], Section 3, Koszmider and T all showed that the answer to Ha jnal and J uhasz’s question is “no”. They also show that the existence of their example is independent o f ZFC. Recall that a top olog ical space is said to be a P -space if each G δ -subset is ope n. It is known that Lindel¨ o f P-s pa ces are Roth b erger spaces [14]. They show in [9], Theorem 4, that : Theorem 10 (Ko szmider-T a ll) . Th e fol lowing is c onsistent r elative to the c onsis- tency of ZF C: CH holds, 2 ℵ 1 > ℵ 2 and every T 2 Lindel¨ of P-sp ac e of c ar dinality ℵ 2 c ontains a c onver gent ω 1 -se quen c e (thus a R othb er ger subsp ac e of c ar dinality ℵ 1 ). And then in Section 3 of [9] they obtain their (consistent) exa mple: Theorem 11 (Koszmider -T all) . It is c onsistent , r elative t o the c onsistency of ZFC, that CH holds and t her e is an unc ountable T 3 -Lindel¨ of P-sp ac e which has no Lin- del¨ of subsp ac e of c ar dinality ℵ 1 . One ma y ask if t he problem of Ha jnal and Juhasz has a solution in certain sub c lasses of the class o f Lindl¨ o f spaces. Koszmider and T all’s results show that even in the class of Rothberger spaces the Ha jnal-Juhas z problem has answer “no” . In the class of Rothberge r spaces with small character the following is known [14]: Theorem 12. If it is c onsistent that ther e is a sup er c omp act c ar dinal, then it is c onsistent tha t 2 ℵ 0 = ℵ 1 , and every unc ountable R othb er ger sp ac e of char acter ≤ ℵ 1 has a R othb er ger subsp ac e of c ar dinality ℵ 1 . 8 MARION SCHEEP ERS F.D. T all co mm unicated to me that th e techniques of this pap er can also b e used to reduce the str ength of the h yp othesis in Theo rem 12 fr o m sup ercompact to measurable. A small additional observ ation converts T all’s re ma rk to the following. Theorem 1 3. Assume ther e is a fr e e ide al J on ω 2 such that TWO has a winning tactic in G + ( J ) . Then every indestructibly Lindel¨ of sp ac e of c ar dinality lar ger than ℵ 1 and of char acter ≤ ℵ 1 has a Ro thb er ger subsp ac e of c ar dinality ℵ 1 . Pro of: If an indestructibly Lindel¨ of spac e has car dinality la rger than ℵ 1 then Theorem 4 implies it has a p oint that is not G δ . If a Lindel¨ of space has character ≤ ℵ 1 and if some element is not a G δ -p oint, then the space has a conv er gent ω 1 - sequence (Theorem 7 in [1]). Such a sequence together with its limit is a Roth b erger subspace.  6. Remarks. (1) Lemma 2 ca n b e stated in greater generality that may b e useful for other applications of these techniques: Lemma 14. L et κ b e a c ar dinal su ch that t her e is a we akly pr e cipitous ide al J ⊂ P ( κ ) . L et X ⊇ κ b e a top olo gic al sp ac e and let F b e a fa mily of G δ subsets of X such that F ⊆ J . Then for e ach F ∈ F and e ach B ∈ J + and e ach se quenc e ( U n ( F ) : n < ω ) of op en neigh b orho o ds of F with F = ∩ n<ω U n ( F ) , t her e i s a C ⊆ B with C ∈ J + and an n such that U n ( F ) ∩ C ∈ J (2) If in a gr ound mo del V we have an ideal J on an ordinal α , then in generic extensions of V let J ∗ denote the ideal on α generated by J . It is of in terest to know which forcings increase 2 ℵ 1 but pr eserve for example the statement: “ There is a σ - complete free ideal J on ω 2 such tha t TW O ha s a winning tactic in G + ( J ∗ )”. This is not preser ved by all ω 1 -complete ω 2 -chain co nditio n partial or ders: In [4] Gorelic shows that for ea ch car dinal num b er κ > ℵ 1 it is co nsistent that CH holds, that 2 ℵ 1 > κ , and there is a T 3 po int s G δ -indestructibly Lindel¨ o f space X of cardinality 2 ℵ 1 . Since the mo del in Section 4 of [2] is a suitable gr ound mo del for Go relic’s construction, Theo rem 4 implies that in the mo del o bta ined by applying Gorelic’s extension to the model from [2], there is no free ideal J on ω 2 such that TWO has a winning tactic in G + ( J ). (3) After learning of the pro of o f Theore m 4, Masa r u Kada informed me that in fact, by known results of F oreman and independently V e li ˇ cko vic, the hypothesis that TWO has a winning strategy in G + ( J ) is sufficient to pr ov e this theor em. W e thank Kada for his kind per mission to include the relev ant r emarks here. F or notation and mo re informa tio n, see [5]: If TWO has a w inning strategy in G + ( J ), the the partially ordered s e t ( J + , ⊆ ) is ω + 1- strategica lly closed, a nd th us (see Corollar y 3.2 in [5]) is strong ly ω 1 -strategica lly closed. In the pr o of o f Theorem 4 use TWO’s winning stra teg y in the game G I ω 1 ( J + ) ins tead of a winning tactic in G + ( J ). Note tha t also the existence o f a free idea l J for which TW O ha s a winning strategy in G + ( J ) is equiconsistent with the existence of a measurable cardinal. References [1] J.E. Baumgartne r and F.D. T all, R efle cting Lindel¨ ofness , T op ology and its Applica tions 122 (2002), 35-49 [2] F. Galvin, T. Jec h and M. Magidor, An ide al game , The Jour nal of Symbolic Logic 43:2 (1978), 284 - 292. MEASURABLE CARDINALS AND THE CARDINALITY OF LINDEL ¨ OF SP ACES 9 [3] F. Galvin and R. T elg´ arsky , Stationary str ate gie s in top olo gica l games , T op ology and it s Applications 22 (1986), 51 - 69. [4] I. Gorelic, The Bair e Cate gory and for cing lar ge Li ndel¨ of sp ac es with p oints G δ , Proceedings of the American Mathematical Society 118 (1993), 603 - 607. [5] T. Ishiu and Y. Y oshinobu, Dir e ctive t r e es and games on p osets , Pro ceedings of the Amer- ican Mathematica l So ciety 130:5 (2001), 1477 - 1485. [6] T. Jec h, Set The ory: The thir d mil lenium ed ition , Spri nger , 2003. [7] T. Jec h, M. Magidor, W. Mitc hell, and K. Prikry , Pr e cipitous Ide als , Journal of Sym b olic Logic 45:1 (1980), 1-8. [8] T. Jech and K. Pr ikry , Ide als over unc ountable sets: A pplic ation of almost disjoint functions and gene ri c ultr ap owers , Memoirs of the Ameri can Mathematical Society 18:214, 1979. [9] P . Koszmider and F.D. T all , A Lindel¨ of sp ac e with no Lindel¨ of subsp ac e of c ar dinality ℵ 1 , Pro ceedings of the American Mathematical Society 130:9 (2002), 2777 - 2787. [10] K. Kunen, Set The ory: An intr o duction to indep e ndenc e pr o ofs , Nor th-Holland 1980. [11] K. Kunen, A. Szymanski and F.D. T all, Bair e irr esolvable sp aces and ide al the ory , Ann. Math. Silesiane 2: 14 (1986), 98 - 107. [12] R. Lav er, O n the c onsistency of Bor el’s c onje ctur e , Acta Mathematicae 137 (1976), 151 - 169. [13] A. L ´ evy and R.M . Solo v ay , Me asur able c ar dinals and the Continuum Hyp othesis , I srael Journal of Mathematics 5 (1967), 234 - 248 [14] M. Scheepers and F.D. T all, Lindel¨ of indestructib i lity, t op olo gic al games and sele ction prin- ciples , s ubmitted. (ArXiv 0902.1944v2) [15] S. Shelah, On some pr oblems in Gener al T op olo gy , Contempora ry Mathema tics 192 (1996), 91 - 101 [16] F.D . T all, On t he c ar dinality of Lindel¨ of sp ac es with p oints G δ , T op ol ogy and it s Appli- cations 63 (1995), 21 - 38.