Transfinite Sequences of Continuous and Baire Class 1 Functions

T ransfinite Sequences of Con tin uous and Baire Class 1 F unctions M´ arto n Elekes and Kennet h Kunen ∗ September 1 8, 2018 Abstract The set of con t in uous or Baire cla ss 1 functions defined on a metric space X is endo w ed with the natural p oin t wise partial order. W e in- v estigate ho w the p ossible lengths of well- ord ered monotone sequ ences (with resp ect to this order) dep end on the space X . In tro duction An y set F of real v alued functions defined on a n arbitrary set X is partially ordered by the p oin t wise order; that is, f ≤ g iff f ( x ) ≤ g ( x ) for all x ∈ X . Then, f < g iff f ≤ g and g 6≤ f ; equiv alently , f ( x ) ≤ g ( x ) for all x ∈ X and f ( x ) < g ( x ) for a t least one x ∈ X . Our aim will b e to in v estigate the p ossible lengths of the increasing or decreasing w ell-ordered sequences o f functions in F with resp ect to this order. A classical theorem (see Kurato wski [7], § 24.I II, Theorem 2 ′ ) asserts that if F is the set of Baire class 1 functions (that is, p oin t wise limits of con tinuous functions) defined on a P olish space X (that is, a complete separable metric space), then there exists a monotone sequence of length ξ in F iff ξ < ω 1 . P . Komj´ ath [5] pro v ed that the corresp onding question concerning Baire class α functions fo r 2 ≤ α < ω 1 is indep ende n t of ZFC . ∗ Partially suppo rted by NSF Gr ant DMS-00978 81. 2000 Mathematics Sub ject Cla ssification: Prima ry 2 6A21; Secondary 03E1 7, 54C3 0. Key w or ds a nd P hr ases: Baire cla ss 1, separable metric spac e, transfinite se quence o f functions. 1 1 SEQUENCES OF CONTINUOUS FUNCTIONS 2 In the presen t pap er we inv estigate what happ ens if w e replace the Polish space X by an arbitrary metric space. Section 1 considers c hains of con tinuous functions. W e sho w that for any metric space X , there exists a chain in C ( X, R ) of order t yp e ξ iff | ξ | ≤ d ( X ). Here, | A | denotes the cardinalit y of the set A , while d ( X ) denotes t he densit y of t he space X , that is d ( X ) = max(min {| D | : D ⊆ X & D = X } , ω ) . In particular, for separable X , every w ell-ordered c ha in has countable length, just as for P olish spaces. Section 2 considers c hains of Baire class 1 functions on separable metric spaces. Here, the situation is en tir ely differen t from the case of P o lish spaces, since on some separable metric spaces, there ar e well-ordered chains of ev ery order t yp e less than ω 2 . F urthermore, the existence of chains o f type ω 2 and longer is indep enden t of ZF C + ¬ CH . Under MA , there are chains of all t yp es less than c + , whereas in the Cohen mo del, all c hains hav e type less than ω 2 . W e note here that instead of examining w ell-ordered sequences, whic h is a classical problem, w e could try to c ha r a cterize all the p ossible o rder t yp es of linearly ordered subsets o f the partially ordered set F . T his problem w as p osed b y M. Laczk o vich , and is considered in detail in [3]. 1 Sequence s of C on tin u ous F unctions Lemma 1.1 F or any top olo gic al sp ac e X : If ther e is a we l l-or der e d se quenc e of len g th ξ in C ( X , R ) , then ξ < d ( X ) + . Pro of. Let { f α : α < ξ } b e a n increasing sequence in C ( X , R ), and let D ⊆ X b e a dense subset of X suc h that d ( X ) = max( | D | , ω ). By con tinuit y , the f α ↾ D a re all distinct; so, for eac h α < ξ , c ho o se a d α ∈ D suc h that f α ( d α ) < f α +1 ( d α ). F or each d ∈ D the set E d = { α : d α = d } is countable, b ecause ev ery w ell-ordered subset o f R is countable. Since ξ = S d ∈ D E d , we ha ve | ξ | ≤ max( | D | , ω ) = d ( X ). The con v erse implication is not true in general. F or example , if X has the coun table c hain condition ( ccc), then ev ery we ll- ordered c hain in C ( X , R ) is coun table (b ecause X × R is also ccc). Ho w ev er, the conv erse is true for metric spaces: Lemma 1.2 If ( X,  ) is any no n-empty metric sp ac e and ≺ is any total or der of the c ar dinal d ( X ) , then ther e i s a chain in C ( X , R ) which is isomorphic to ≺ . 2 SEQUENCES OF BAIRE CLASS 1 FUNCTIONS 3 Pro of. First, note that ev ery countable to tal order is em b eddable in R , so if d ( X ) = ω , then the result follow s trivially using constan t functions. In particular, we ma y assume that X is infinite, and then fix D ⊆ X whic h is dense and o f size d ( X ). F or each n ∈ ω , let D n b e a subset of D whic h is maximal with resp ect to the prop ert y ∀ d, e ∈ D n [ d 6 = e →  ( d, e ) ≥ 2 2 − n ]. Then S n D n is also dense, so w e may assume that S n D n = D . W e may also assume that ≺ is a to t al order of the set D . N ow, w e shall pro duce f d ∈ C ( X, R ) for d ∈ D suc h that f d < f e whenev er d ≺ e . F or eac h n , if c ∈ D n , define ϕ n c ( x ) = max(0 , 2 − n −  ( x, c )). F or eac h d ∈ D , let ψ n d = P { ϕ n c : c ∈ D n & c ≺ d } . Since ev ery x ∈ X has a neighbor ho o d on whic h all but at mo st one of the ϕ n c v anish, we hav e ψ n d ∈ C ( X , [0 , 2 − n ]), and ψ n d ≤ ψ n e whenev er d ≺ e . Thus , if w e let f d = P n<ω ψ n d , w e ha ve f d ∈ C ( X , [0 , 2]), and f d ≤ f e whenev er d ≺ e . But also, if d ∈ D n and d ≺ e , then ψ n d ( d ) = 0 < 2 − n = ψ n e ( d ), so actually f d < f e whenev er d ≺ e . Putting these lemmas together, w e hav e: Theorem 1.3 L et ( X,  ) b e a metric sp ac e. Then ther e e x ists a we l l-or der e d se quenc e of length ξ in C ( X , R ) iff ξ < d ( X ) + . Corollary 1.4 A metric sp ac e ( X ,  ) is sep ar able iff eve ry we l l-or der e d se- quenc e in C ( X , R ) is c ountable. 2 Sequence s of Bair e Class 1 F unctions If we replace con tinuous functions by Baire class 1 functions, then Corollar y 1.4 b ecomes false, since on some separable metric spaces, we can g et w ell-ordered sequence s of ev ery type less than ω 2 . T o pro v e this, w e shall apply some basic facts ab out ⊂ ∗ on P ( ω ). As usual, for x, y ⊆ ω , w e sa y that x ⊆ ∗ y iff x \ y is finite. Then x ⊂ ∗ y iff x \ y is finite and y \ x is infinite. This ⊂ ∗ partially orders P ( ω ). Lemma 2.1 If X ⊂ P ( ω ) is a ch ain in the or der ⊂ ∗ , then on X (viewe d as a subset of the Cantor s e t 2 ω ∼ = P ( ω ) ), ther e is a chain of Bair e class 1 functions which is isomorphi c to ( X , ⊂ ∗ ) . Pro of. Note that for each x ∈ X , { y ∈ X : y ⊆ ∗ x } = [ m ∈ ω { y ∈ X : ∀ n ≥ m [ y ( n ) ≤ x ( n )] } , 2 SEQUENCES OF BAIRE CLASS 1 FUNCTIONS 4 whic h is an F σ set in X . Lik ewise, the sets { y ∈ X : y ⊇ ∗ x } , { y ∈ X : y ⊂ ∗ x } , and { y ∈ X : y ⊃ ∗ x } , are all F σ sets in X , a nd hence also G δ sets. It follows that if f x : X → { 0 , 1 } is the c har acteristic function of { y ∈ X : y ⊂ ∗ x } , then f x : X → R is a Baire class 1 function. Then, { f x : x ∈ X } is the required c hain. Lemma 2.2 F or a ny infinite c ar dinal κ , supp ose that ( P ( ω ) , ⊂ ∗ ) c ontains a chain { x α : α < κ } (i.e., α < β → x α ⊂ ∗ x β } ). Then ( P ( ω ) , ⊂ ∗ ) c ontains a chain X of size κ such that every or dinal ξ < κ + is em b e ddable into X . Pro of. Let S = S 1 ≤ n<ω κ n . F or s = ( α 1 , . . . , α n − 1 , α n ) ∈ S , let s + = ( α 1 , . . . , α n − 1 , α n + 1). Starting with the x ( α ) = x α , choose x s ∈ P ( ω ) by induction o n length( s ) so that x s = x s ⌢ 0 ⊂ ∗ x s ⌢ α ⊂ ∗ x s ⌢ β ⊂ ∗ x s + whenev er s ∈ S and 0 < α < β < κ . L et X = { x s : s ∈ S } . Then, whenev er x, y ∈ X with x ⊂ ∗ y , the ordinal κ is em b eddable in ( x, y ) = { z ∈ X : x ⊂ ∗ z ⊂ ∗ y } . F rom this, one easily prov es b y induction on ξ < κ + (using cf ( ξ ) ≤ κ ) that ξ is embeddable in each suc h in terv al ( x, y ). Since P ( ω ) certainly contains a c hain of ty p e ω 1 , these tw o lemmas yield: Theorem 2.3 Ther e is a se p ar able metric sp ac e X on whic h, for every ξ < ω 2 , ther e is a wel l-or der e d chain of le n gth ξ of B air e class 1 functions. Under C H , this is b est possible, since there will b e only 2 ω = ω 1 Baire class 1 functions on a separable metric space, so there could not b e a c hain of length ω 2 . Under ¬ CH , the existenc e of longer c hains of Baire class 1 functions dep ends on the mo del of set theory . It is consisten t with c = 2 ω b eing arbitrarily large that there is a chain in ( P ( ω ) , ⊂ ∗ ) of type c ; for example, this is true under MA (see [2]). In this case, there will b e a separable X with w ell-ordered chains of all lengths less than c + . Ho w eve r, in the Cohen mo del, where c can also b e made arbitrarily large, w e nev er get c hains of t yp e ω 2 . W e shall pro v e this b y using the following lemma, whic h relates it to the rectangle problem: Lemma 2.4 Supp ose that ther e is a sep ar able m e tric sp ac e Y with an ω 2 -chain of Bor el subsets, { B α : α < ω 2 } (so, α < β → B α $ B β ). Then in ω 2 × ω 2 , the wel l-or der r elation < is in the σ -alg e br a gener ate d by the set of al l r e ctangles, { S × T : S, T ∈ P ( ω 2 ) } . 2 SEQUENCES OF BAIRE CLASS 1 FUNCTIONS 5 Pro of. Each B α has some coun table Bor el rank. Since there are only ω 1 ranks, w e may , by passing to a subsequence, assume that the ra nks are b ounded. Sa y , eac h B α is a Σ 0 µ set fo r some fixed µ < ω 1 . Let J = ω ω , and let A ⊆ Y × J b e a unive rsal Σ 0 µ set; that is, A is Σ 0 µ in Y × J and eve r y Σ 0 µ subset of Y is of the form A j = { y : ( y , j ) ∈ A } for some j ∈ J (see [7], § 31). Now , for α, β < ω 2 , fix y α ∈ B α +1 \ B α , and fix j β ∈ J suc h that A j β = B β . Then α < β iff ( y α , j β ) ∈ A . Th us, { ( y α , j β ) : α < β < ω 2 } is a Borel subset of { y α : α < ω 2 } × { j β : β < ω 2 } , and is hence in the σ - algebra generated b y op en rectangles, so < , as a subset of ω 2 × ω 2 , is in the σ -algebra generated b y rectangles. Theorem 2.5 Assume that the wel l-or der r elation < on ω 2 is not in the σ - algebr a gener ate d by the set of al l r e ctangles. Th en no sep ar able m etric sp ac e c an have a chain o f leng th ω 2 of B a ir e class 1 functions. Pro of. Suppose t ha t { f α : α < ω 2 } is a c hain of Baire class one functions on the separable metric space X . Let B α = { ( x, r ) ∈ X × R : r ≤ f α ( x ) } . Then the B α form an ω 2 -c hain of Borel subsets of the separable metric space X × R , so w e ha ve a con tradiction b y Lemma 2.4. Finally , w e p oint out that t he hypothesis of this theorem is consisten t, since it holds in the extension V [ G ] f o rmed by adding ≥ ω 2 Cohen r eals to a ground mo del V whic h satisfies CH . This fact w a s first prov ed in [6]. It also follows from the more general principle H P 2 ( ω 2 ) o f Brendle, F uc hino, and Soukup [1]. They define this principle, prov e that it holds in Cohen extensions (and in a n um b er of other forcing extensions), and sho w the follo wing: Lemma 2.6 H P 2 ( κ ) imp l i es that if R is any r elation on P ( ω ) w h ich is first- or der definable over H ( ω 1 ) fr om a fixe d el e ment of H ( ω 1 ) , then ther e is no X ⊆ P ( ω ) such that ( X ; R ) is is o morphic to ( κ ; < ) . These matters are also discussed in [4], whic h indicates how suc h statemen ts are ve rified in Cohen extensions. Here, H ( ω 1 ) denotes the set of hereditarily coun table sets. Lemma 2.7 H P 2 ( ω 2 ) imp lies that in ω 2 × ω 2 , the w el l-or der r elation < is not in the σ -alge br a gener ate d by the set of al l r e ctangles, { S × T : S, T ∈ P ( ω 2 ) } . Pro of. Supp ose that < w ere in this σ -algebra. Then we w ould ha v e fixed K n ⊆ ω 2 for n < ω such that < is in the σ -algebra generated by all the K m × K n . REFERENCES 6 F or each α , let u α = { n ∈ ω : α ∈ K n } . There is then a form ula ϕ ( x, y , z ) and a fixed w ∈ H ( ω 1 ) such that for all α, β < ω 2 , α < β iff H ( ω 1 ) | = ϕ ( u α , u β , w ); here, w enco des the par t icular countable b o olean comb ination used to get < from the K n . No w, if X = { u α : α < ω 2 } , then ϕ defines a relation R on H ( ω 1 ) suc h that ( X ; R ) is isomorphic to ( ω 2 ; < ), con tradicting Lemma 2.6. References [1] J. Brendle, S. F uc hino, and L. Soukup, Coloring ordinals b y reals, to app ear. [2] E. K. v an Douw en, The in tegers and top ology , in Handb o ok of Set- Theoretic T op ology , North-Holland, Amsterdam, 1984, pp. 111-167 . [3] M. Elek es, Linearly ordered families of Baire 1 functions, Real Analysis Exc hange , to app ear. [4] I. Juh´ asz and K. Kunen, The p o w er set o f ω , F undamen ta Mathematicae , V ol 170 (20 0 1), 257- 265. [5] P . K o mj´ ath, Ordered families of Baire- 2 -functions, Real Analysis Ex- c hange , V ol 15 (1 9 89-90), 442-444. [6] K. Kunen, Inaccessibilit y Prop erties o f Cardinals , Do ctoral Dissertation, Stanford, 1968. [7] K. Kura to wski, T op olo gy , V o l. 1 , Academic Press, 1966. Dep ar tment of Anal ysis, E ¨ otv ¨ os Lor ´ and University, Budapest, P ´ azm ´ any P ´ eter s ´ et ´ any 1/c, 1117, Hungar y Email a ddr ess : emarc i@cs.elte.hu Dep ar tment of Ma thema tics, Univers ity of Wisconsin, Madison, WI 53706, US A Email a ddr ess : kunen @math.wisc.edu URL: http://www.ma th.wisc.edu / ~kunen