Algebrability, non-linear properties, and special functions
Artur Bartoszewicz, Szymon Glab, Daniel Pellegrino, Juan B. Seoane-Sepúlveda
aa r X i v : . [ m a t h . F A ] S e p ALGEBRABILITY, NON-LINEAR PROPERTIES, AND SPECIAL FUNCTIONS
ARTUR BARTOSZEWICZ * , SZYMON G LA¸ B * , DANIEL PELLEGRINO,AND JUAN B. SEOANE-SEP ´ULVEDA ** Abstract.
We construct uncountably generated algebras inside the following sets of special func-tions: (i) Sierpi´nski-Zygmund functions. (ii) Perfectly everywhere surjective functions. (iii) Nowherecontinuous Darboux functions. All conclusions obtained in this paper are improvements of somealready known results. Preliminaries and background
This paper is a contribution to the very recent trend, in mathematical analysis, of the searchfor large algebraic structures (linear spaces or algebras) enjoying what one could call “ special ”properties. As it has become a usual notion nowadays, given a certain property we say that thesubset M of a topological vector space X which satisfies it is µ -lineable (respectively, µ -spaceable )if M ∪ { } contains a vector space (respectively, closed vector space) of dimension µ (finite orinfinite). If M contains an infinite-dimensional (closed) vector space, it shall be simply called lineable ( respectively spaceable) for short. These notions of lineability and algebrability were coinedby V. I. Gurariy in the early 2000’s and first introduced in [4, 40].In the recent years many examples of vector spaces of functions on R or C enjoying certain specialproperties have been constructed. Although these latter notions of lineability and spaceability didnot appear until recently, the origins of this theory date back to 1966, when a famous exampledue to V. I. Gurariy ([30, 31]) showed that there exists an infinite dimensional linear space everynon-zero element of which is a continuous nowhere differentiable function on C [0 , R is 2 c -lineable (where c denotes thecardinality of R ) and that the set of differentiable functions on R which are nowhere monotone islineable in C ( R ). These behaviors occur, sometimes, in particularly interesting ways. For example,in [32], Hencl showed that any separable Banach space is isometrically isomorphic to a subspaceof C [0 ,
1] whose non-zero elements are nowhere approximately differentiable and nowhere H¨older.We refer the interested reader to [2, 3, 10–13, 15, 16, 22, 25, 27, 37] for a wider range of results in thistopic of lineability and spaceability.Of course, one could go further and not just consider linear spaces but, instead, larger or morecomplex structures. For instance, in [1] the authors showed that there exists and uncountablygenerated algebra every non-zero element of which is an everywhere surjective function on C , and Mathematics Subject Classification.
Primary: 15A03; Secondary: 28A20, 46J10.
Key words and phrases. lineability; spaceability; algebrability; perfectly everywhere surjective functions;Sierpi´nski-Zygmund functions. * Supported by the Polish Ministry of Science and Higher Education Grant No. N N201 414939 (2010-2013). ** Supported by the Spanish Ministry of Science and Innovation, grant MTM2009-07848. in [5] it was shown that, if E ⊂ T is a set of measure zero, and if F ( T ) denotes the subset of C ( T ) of continuous functions whose Fourier series expansion diverges at every point of E , then F ( T ) contains an infinitely generated and dense subalgebra. These previous examples motivatedthe notion of algebrability . This notion was first introduced in [5, 6] and here we give an slightlysimplified version of it, that shall be sufficient for our purposes. Definition 1.1 ([5, 6]) . Let L be an algebra. A set A ⊂ L is said to be β -algebrable if there existsan algebra B so that B ⊂ A ∪{ } and card( Z ) = β , where β is a cardinal number and Z is a minimalsystem of generators of B . Here, by Z = { z α : α ∈ Λ } is a minimal system of generators of B , wemean that B = A ( Z ) is the algebra generated by Z, and for every α ∈ Λ , z α / ∈ A ( Z \ { z α } ) . Wealso say that A is algebrable if A is β -algebrable for β infinite. Remark . (i) Notice that if Z is a minimal infinite system of generators of B , then A ( Z ′ ) = B for any Z ′ ⊂ B such that card( Z ′ ) < card( Z ). The result is not true for finite systemsof generators: Take X = C with coordinate-wise multiplication. X is a Banach algebrawith unit (1 , { (1 , , (0 , } is a minimal system of generators of X . However, X is also single generated by u = (1 , i ): Consider P : X → X, P ( s, t ) = ( s , t ). Notethat P ( u ) = (1 , −
1) and so we get i ( u − P ( u )) = (0 , ∈ X. Similarly, we also have(1 , ∈ X. (ii) Of course, notice that algebrability implies lineability, and the converse (in general) doesnot hold. For instance (and as it was proved in [28]), given any unbounded interval I , theset of Riemann-integrable functions on I that are not Lebesgue-integrable is lineable andnot algebrable.There also exists an strengthening of the notion of algebrability (see [9]). Given a cardinality κ ,we say that A is a κ -generated free algebra if there exists a subset X = { x α : α < κ } of A such thatany function f from X to some algebra A ′ can be uniquely extended to a homomorphism from A into A ′ . Then X is called a set of free generators of the algebra A . A subset X = { x α : α < κ } ofa commutative algebra B generates a free sub-algebra A if and only if for each polynomial P andany x α , x α , . . . , x α n we have P ( x α , x α , . . . , x α n ) = 0 if and only if P = 0. Also, let us recall that X = { x α : α < κ } ⊂ E is a set of free generators of a free algebra A ⊂ E if and only if the set b X ofelements of the form x k α x k α · · · x k n α n is linearly independent and all linear combinations of elementsfrom b X are in E ∪ { } . With this at hand, let us recall the following definition (see [9]). Definition 1.3.
We say that a subset E of a commutative linear algebra B is strongly κ -algebrableif there exists a κ -generated free algebra A contained in E ∪ { } .In general, there are subsets of linear algebras which are algebrable but not strongly algebrable(for instance, c is algebrable in c but it is not strongly 1-algebrable, as it is shown in [9]).This paper is mainly devoted to the thorough study of the following recently considered classes offunctions: • Sierpi´nski-Zygmund functions. • Perfectly everywhere surjective functions. • Nowhere continuous Darboux functions (nowhere continuous functions mapping connectedsets to connected sets).These latter classes have been lately considered in [21, 24–26], where the authors proved theexistence of infinite dimensional linear spaces inside each of the previous classes. In particular, in[24, Theorem 5.6] it was proved that the set of Sierpi´nski-Zygmund functions is c + -lineable. In LGEBRABILITY, NON-LINEAR PROPERTIES, AND SPECIAL FUNCTIONS 3 [24, Theorem 2.6] the authors showed the 2 c -lineability of the set of perfectly everywhere surjectivefunctions, and in [25, 26] similar results were obtained on the class of nowhere continuous functionsmapping connected sets to connected sets. The aim of this paper is to improve all the previouslymentioned results by obtaining algebrability and strong algebrability where just lineability waspreviously reached. In particular, and among other results, we shall prove that there exists anuncountably generated algebra every non-zero element of which is a nowhere continuous Darbouxfunction (Theorem 4.6) and that both, the sets of Sierpi´nski-Zygmund functions (Theorem 2.6)and the set of perfectly everywhere surjective functions (Theorem 3.5), are (respectively) strongly c + -algebrable and strongly 2 c -algebrable.We shall use standard set theoretical notation. As usual, ω shall denote the cardinality of N and c shall denote the continuum. Also, we shall identify each cardinal number with the first ordinalof the same cardinality. Then a cardinal κ is equal to the set of all ordinals less than κ , denoted κ = { ξ : ξ < κ } . The rest of the notation shall be rather usual. Classical Real Analysis and AbstractAlgebra techniques together with Set Theory and Cardinal Theory shall be used throughout thepaper. 2. Sierpi´nski-Zygmund functions
This section shall be devoted to a type of function that first appeared in the early 1920’s (see[41]) and due to W. Sierpi´nski and A. Zygmund. A clear consequence of the classic Luzin’s Theoremis that, for every measurable function f : R → R , there exists a measurable set S ⊂ R , of infinitemeasure, such that f | S is continuous. On the other hand, and given any arbitrary function f : R → R , is it possible to find a dense set S ⊂ R such that f | S is continuous? The answer to this questionwas given by Blumberg ([14], 1922): Theorem 2.1 (Blumberg) . Let f : R → R be an arbitrary function. There exists a dense subset S ⊂ R such that the function f | S is continuous. Checking the proof of Blumberg’s theorem (see, [34, p. 154]) it can be noticed that the set S above is countable. Naturally, we could wonder whether we can choose the set S in Blumberg’stheorem to be uncountable. This was partially answered (in the negative) by W. Sierpi´nski and A.Zygmund a year later in [41] (see also [34, pp. 165-166] for a modern reference). Theorem 2.2 (Sierpi´nski-Zygmund) . There exists a function f : R → R such that, for any set Z ⊂ R of cardinality the continuum, the restriction f | Z is not a Borel map (and, in particular, notcontinuous). Notice that, if the Continuum Hypothesis holds, the restriction of this function to any uncount-able set is not continuous. The Continuum Hypothesis is necessary in this frame. Shinoda ([39])showed that if both Martin’s Axiom and the negation of the Continuum Hypothesis hold, then forevery f : R → R there exists an uncountable set Z ⊂ R such that f | Z is continuous. Following thenotation from [24], we say that a function f : R → R is a Sierpi´nski-Zygmund function if it satisfiesthe condition in Sierpi´nski-Zygmund’s Theorem and the set of these functions is denoted by SZ ( R ) = { f : R → R : f is a Sierpi´nski-Zygmund function } . In [24, Theorems 5.6 and 5.10] it was proved that the set SZ ( R ) is c + -lineable and, also, c -algebrable. As a consequence, assuming that c + = 2 c (which follows, for instance, from theGeneralized Continuum Hypothesis or GCH), SZ ( R ) is also 2 c -lineable. Up to today it was notknown whether any additional set-theoretical assumptions is needed or not in order to show the A. BARTOSZEWICZ, S. G LA¸ B, D. PELLEGRINO, AND J. B. SEOANE-SEP ´ULVEDA c -lineability of SZ ( R ). In Theorem 2.6 we shall improve this latter results from [24] by provid-ing a new technique that shall show that the set of all Sierpi´nski-Zygmund functions is actually κ -algebrable for some c + ≤ κ ≤ c .Let us recall that, besides its intrinsic mathematical interest, this class of Sierpi´nski-Zygmundfunctions has attracted the interest of many mathematicians, especially in the area of Real Analysis,since its appearance at the beginning of the 20th century (see, e.g. [8, 17, 18, 38]). In [8] it wasshown that there exists a Darboux function that is also in SZ ( R ) and that there exists a model ofZFC in which there are no such functions.Let us begin by giving certain results that shall be needed throughout this section. Lemma 2.3.
Let P be a non-zero polynomial in n + 1 variables. Then the set { x : ∀ y ∈ R n , P ( x, y ) = 0 } is finite.Proof. Suppose that the set { x : ∀ y ∈ R n P ( x, y ) = 0 } is infinite. Let y ∈ R n . Then the polynomial x P ( x, y ) in one variable has infinitely many zeros, and therefore it vanishes for every x . Hence P ≡ (cid:3) Lemma 2.4.
Let P be a family of polynomials of cardinality less than c . Then there exists a set Y = { y ξ : ξ < c } such that P ( y ξ , y ξ , . . . , y ξ n ) = 0 for any n ≥ , any polynomial P ∈ P in n variables and any distinct ordinals ξ i < c .Proof. Let λ = |P| . By P n we denote the set of polynomials from P in n variables. Without lossof generality we may assume that P is closed under taking permutations of variables, i.e. for any n , P ∈ P n and any permutation σ ∈ S n , the polynomial Q defined by Q ( x , x , . . . , x n ) = P ( x σ (1) , x σ (2) , . . . , x σ ( n ) )is in P .We define { y ξ : ξ < c } by induction. Suppose that we have already defined Y α = { y ξ : ξ < α } such that(1) ∀ n < min { ω, α + 1 }∀ P ∈ P n ∀ ξ < ξ < . . . < ξ n < α ( P ( y ξ , y ξ , . . . , y ξ n ) = 0)(2) ∀ n ∀ m < min { n, α +1 }∀ P ∈ P n ∀ ξ < ξ < . . . < ξ m < α ∃ y ∈ R n − m ( P ( y ξ , y ξ , . . . , y ξ m , y ) =0).We find y α / ∈ Y α such that(1 ′ ) ∀ n < min { ω, α + 1 }∀ P ∈ P n +1 ∀ ξ < ξ < . . . < ξ n < α ( P ( y ξ , y ξ , . . . , y ξ n , y α ) = 0)(2 ′ ) ∀ n ∀ m < min { n, α +1 }∀ P ∈ P n +1 ∀ ξ < ξ < . . . < ξ m < α ∃ y ∈ R n − m ( P ( y ξ , y ξ , . . . , y ξ m , y α , y ) =0).Note that, by (2), for a given P ∈ P n +1 and ξ < . . . < ξ n < α the set { x : P ( y ξ , . . . , y ξ n , x ) = 0 } is finite. Therefore the set W := { x : ∃ n < min { ω, α + 1 }∃ P ∈ P n +1 ∃ ξ < . . . < ξ n < α ( P ( y ξ , . . . , y ξ n , x )) = 0 } is of cardinality less than max { ω, α, λ } < c .Now, by Lemma 2.3 and (2), we have that for a given P ∈ P n +1 , m < min { n, α + 1 } and ξ < . . . < ξ m < α the set { x : P ( y ξ , . . . , y ξ m , x, y ) = 0 for all y ∈ R n − m } is finite. Therefore theset X := { x : ∃ n ∃ m < min { n, α + 1 }∃ P ∈ P n +1 ∃ ξ < . . . < ξ n < α ∀ y ∈ R n − m ( P ( y ξ , . . . , y ξ n , x, y )) = 0 } LGEBRABILITY, NON-LINEAR PROPERTIES, AND SPECIAL FUNCTIONS 5 is of cardinality less than max { ω, α, λ } < c . Since Y α is of cardinality less than c we can find y α which is not in Y α ∪ W ∪ X . This finishes the inductive proof. (cid:3) Lemma 2.5.
Let P be a family of non-zero real polynomials with no constant term and let X bea subset of R both of cardinality less than c . Then there exists set Y = { y ξ : ξ < c } such that P ( y ξ , y ξ , . . . , y ξ n ) / ∈ X for any n , any polynomial P ∈ P and any distinct ordinals ξ i < c .Proof. Consider the family of polynomials P ′ = { P − x : P ∈ P , x ∈ X } . Then using Lemma 2.4for P ′ we easily obtain the assertion. (cid:3) Before stating and proving the main result of this section, let us recall the set theoretical notionof almost disjoint sets . Although this notion admits several variants depending on the frameworkwe are working on, given A and B any two subsets of R of cardinality c we shall say that A and B are almost disjoint if | A ∩ B | < c (see, e.g. [19, Section 1]). Let us recall that the existence of2 c almost disjoint subsets of the real line each of them of cardinality c follows, for instance, fromMartin’s Axiom (MA) and CH (see, e.g. [35]). It is also known that, in general (see [19]), thereexists an almost disjoint family of cardinality κ + on κ .Now, with this at hand (and in the usual framework of ZFC) it is time to state and prove themain result of this section. Theorem 2.6.
The set of Sierpi´nski-Zygmund functions is strongly κ -algebrable, provided thereexists a family of κ almost disjoint subsets of c .Proof. Let { g α : α < c } be a well-ordering of all Borel functions and write R = { x α : α < c } .Finally, let { P α : α < c } denote the set of all non-zero polynomials without constant term. Define,inductively, { Y α : α < c } a family of subsets of R each of which has cardinality c . At the stage α we use Lemma 2.5 for X := { g λ ( x α ) : λ ≤ α } and P := { P β : β ≤ α } to define Y α . Next, let Y α = { y αξ : ξ < c } and define Y = Q α< c Y α . Let { N ζ : ζ < κ } be a set of almost disjoint subsetsof c each of cardinality c . For any ζ < κ let { ζ ( ξ ) : ξ < c } be an increasing enumeration of N ζ anddefine f ζ : R → R by f ζ ( x α ) = y αζ ( α ) . It suffices to show that { f ζ : ζ < κ } is a set of free generatorsof an algebra consisting of Sierpi´nski-Zygmund functions.. Let ζ < ζ < . . . < ζ n < κ , P β be apolynomial in n variables, g γ be a Borel function and Z be any subset of R of cardinality c . Since { N ζ : ζ < κ } is an almost disjoint family of c , there is ξ < c such that N ζ , N ζ , . . . , N ζ n are disjointabove ξ (i.e. N ζ i ∩ ( ξ, c ) are pairwise disjoint, where ( ξ, c ) is a set consisting of all ordinals between ξ and c ). Since Z is of cardinality c , there is α < c with α > max { β, γ, ξ } and x α ∈ Z . Since α is greater than ξ , then f ζ ( x α ) , f ζ ( x α ) , . . . , f ζ n ( x α ) are distinct points of Y α . Since α is greaterthan β and γ , by construction P β ( f ζ , f ζ , . . . , f ζ n ) differs from g γ at the point x α ∈ Z . Therefore P β ( f ζ , f ζ , . . . , f ζ n ) is a Sierpi´nski-Zygmund function and the result follows. (cid:3) Of course, we have the following consequences.
Corollary 2.7.
If one of the following set-theoretical assumption holds • Martin’s Axiom, or • CH or, • c + = 2 c ,then the set of Sierpi´nski-Zygmund functions is c -algebrable. Also, and from the previous corollary, we can obtain (as particular cases) [24, Theorem 5.6] and[24, Theorem 5.10]. The following natural question now follows:
A. BARTOSZEWICZ, S. G LA¸ B, D. PELLEGRINO, AND J. B. SEOANE-SEP ´ULVEDA
Question 2.8.
Is it necessary to add any additional hypothesis to ZFC in order to obtain 2 c -algebrability (or even 2 c -lineability) of SZ ( R )?3. Perfectly everywhere surjective functions
In the last years many authors have been interested in the study of different degrees of surjectivity of functions in R (see, e.g. [1, 4, 6, 21, 24]). Lebesgue [29, 36] was probably the first to show anexample of a function f : R → R with the property that on every non-trivial interval I , f ( I ) = R .In [4] the authors proved that the set of these so called everywhere surjective functions is 2 c -lineable,which is the best possible result in terms of dimension. One could think that, in terms of surjectivity,these everywhere surjective functions are somewhat the most pathological . Well, actually (and aswe shall see in this and the following section) even more pathological surjective functions exist.For instance, there exists a function f : R → R such that for every perfect set P and for every r ∈ R the set { x ∈ P : f ( x ) = r } has cardinality c (see [20, pp. 124-125] and [24, Example 2.1]).The latter functions are called perfectly everywhere surjective . A perfectly everywhere surjectivefunction actually attains every real value c times on any perfect set. Clearly, if f is perfectlyeverywhere surjective then f is everywhere surjective and the converse is false (see [24, Example2.2]). In this section we shall consider complex functions f : C → C and we shall say that f isperfectly everywhere surjective provided f ( P ) = C for every perfect subset P of the complex plane.Recently, it was proved that: Theorem 3.1 ([24]) . The set of perfectly everywhere surjective functions on C is ω -algebrable. Here, our aim is to improve the above theorem by proving that the set of perfectly everywheresurjective functions on C is, actually, 2 c -algebrable, which (in terms of cardinalities and dimensions)is the best possible result. From now on we shall denote PES ( C ) = { f : C → C : f is perfectly everywhere surjective } . Let us begin by recalling that a subset B of C is called a Bernstein set if B and its complement B c meet every perfect subset of C . Since every perfect set P contains c many perfect subsets, then P ∩ B has cardinality c . Note that any polynomial Q : C → C is an entire function, and thus if f isperfectly everywhere surjective then so is Q ( f ); in particular this means that f is a free generatorin the algebra C C with the usual point-wise addition and product. The following lemma can befound in [9]. Lemma 3.2.
Let A be a commutative algebra with unit. Let x ∈ A be a free generator (i.e. x, x , x , . . . are linearly independent) and let P = 0 be a polynomial in n variables without non-zeroconstant term. Then at least one of the elements P ( x, , . . . , , P (1 , x, , . . . , , . . . , P (1 , . . . , , x ) is non-zero. The following proposition shall be necessary in order to prove the main result of this section,and its proof follows the same spirit of that in [24, Example 2.1].
Proposition 3.3.
Let B ⊂ C be a Bernstein set. There exist f ∈ PES ( C ) which is identically 1on the complement of B .Proof. First of all let us recall that if P is a perfect set then P ∩ B has cardinality c . Now, thecardinality of the collection of pairs ( P ∩ B, y ) where P is a perfect set and y ∈ C is c . Let { A α : α < c } be a well-ordering of this set and notice that each perfect set P occurs c times at afirst part of a pair ( P ∩ B, y ). LGEBRABILITY, NON-LINEAR PROPERTIES, AND SPECIAL FUNCTIONS 7
Assume that A = ( P ∩ B, y ), choose x ∈ P ∩ B and define f ( x ) = y . Let us assume nowthat, for each β < α , A β = ( P β ∩ B, y β ) and that a point x β ∈ P β ∩ B has been chosen in such away that for every γ < β , x β = x γ and with f ( x β ) = y β . Next, let A α = ( P α ∩ B, y α ). Since thecardinality of the set { x β : β < α } is less than c , there exists x α ∈ ( P α ∩ B ) \ { x β : β < α } . Now,define f ( x α ) = y α . Thus, for each α < c , a point has been chosen so that for β < α , x α = x β and f ( x α ) = y α . If x / ∈ { x α : α < c } , we define f ( x ) = 1. By doing this, f is now defined on C .Now, if P is any perfect set and y ∈ C , then there exists α < c such that ( P ∩ B, y ) = A α .Therefore there exists x = x α ∈ P ∩ B with f ( x ) = y , and thus f ( P ) ⊃ f ( P ∩ B ) = C . Also, andby construction, it can be checked that f is identically 1 on B c . (cid:3) Before carrying on, let us first recall the notion of independence in abstract set theory (see, e.g.[7]).
Definition 3.4.
A family ( A s ) of subsets of κ (or any fixed set of cardinality κ ) is called independentif the set A i s ∩ . . . ∩ A i n s n has cardinality κ for any n ∈ N , distinct s , . . . , s n and any i k ∈ { , } ,where we denote A = A and A stands for the complement of A for any set A .From a general result due to Balcar and Franˇek (see [7]), it follows that for any set of cardinality κ , there exists an independent family of its subsets of cardinality the largest possible, i.e. 2 κ . Thisshall be used in the main result of this section, that we state and prove now. Theorem 3.5.
The set
PES ( C ) is strongly c -algebrable.Proof. Let { A ζ : ζ < c } be an independent family of subsets of c . Let { B α : α < c } be a family ofpairwise disjoint Bernstein subsets of C . Let g α be a perfectly everywhere surjective function as inProposition 3.3 (for B α ). Let us define f ζ : C → C as follows: f ζ ( x ) = (cid:26) g α ( x ) if x ∈ B α and α ∈ A ζ , . We shall prove that { f ζ : ζ < c } is a family of free generators. Let ζ < ζ < . . . < ζ n < c andlet P be a polynomial in n variables. It suffices to show that P ( f ζ , f ζ , . . . , f ζ n ) is in PES ( C ).Since { A ζ : ζ < c } is independent, there is α ∈ A ζ \ S ni =2 A ζ i . Then P ( f ζ , f ζ , . . . , f ζ n ) | B α = P ( g α | B α , , . . . , g α is a free generator. Therefore, byLemma 3.2, one of the elements P ( f ζ , , . . . , P (1 , f ζ , , . . . , . . . , P (1 , . . . , , f ζ ) is non-zeroon B α , and therefore is in PES ( C ). Without loss of generality we may assume that P (1 , . . . , , f ζ )is non-zero on B α .Now, we shall show that P (1 , . . . , , f ζ n ) is non-zero and, therefore belongs to PES ( C ). Since { A ζ : ζ < c } is independent, there exists β ∈ A ζ n \ S n − i =1 A ζ i . Then f ζ n is equal to g β over B β . Notice also that the fibres g − α ( x ) and g − β ( x ) have cardinality c for every x ∈ C . Thus,there is a bijection h : B α ↔ B β such that h maps g − α ( x ) onto g − β ( x ) for every x ∈ C . Then f ζ | B α = f ζ n | B β ◦ h . Hence P (1 , . . . , , f ζ | B α ) = P (1 , . . . , , f ζ n | B β ◦ h )is non-zero. Thus, P (1 , . . . , , f ζ n | B β ) is non-zero and the result follows. (cid:3) Nowhere continuous Darboux functions
As it is known (at an undergraduate level) if a function f ∈ R R is continuous, then it mapsconnected sets to connected sets (in other words, it is a Darboux function). Very recently (see[22] and [26, Theorem 1]) it was constructed a 2 c -dimensional linear space of functions in R R every A. BARTOSZEWICZ, S. G LA¸ B, D. PELLEGRINO, AND J. B. SEOANE-SEP ´ULVEDA non-zero element of which is nowhere continuous and Darboux. In this section we are interested in,again, moving from lineability to algebrability, trying to improve the result just mentioned. In orderto do that, we shall need to define several notions that shall be of help throughout this section.Of course, and although there is no need in saying it here, if f ∈ PES ( R ) then f is nowherecontinuous. As we mentioned in the previous section, a function f ∈ R R is called everywheresurjective (denoted f ∈ ES ( R ), [4]) if for every non-void interval I of R we have f ( I ) = R . It isclear that PES ( R ) ⊂ ES ( R ) (see [24]).If f ∈ R R , we say that f ∈ J ( R ) ( f is a Jones function) if for every closed set K ⊂ R withuncountable projection on the x -axis, we have f ∩ K = ∅ (see [33]). As it was proved in [25], thefollowing holds J ( R ) ⊂ PES ( R ) ⊂ ES ( R ) . Interestingly enough, it is known ([24, 25]) that neither perfectly everywhere surjective functionsnor Jones functions are measurable and, in contrast with Section 2 in this paper, we also have that([25, Corollary 3.8]) SZ ( R ) ∩ J ( R ) = ∅ .Throughout this section, we shall identify a function from R to R with its graphic, that is, weshall consider that function as a subset of R . Nevertheless, sometimes it is more convenient theopposite procedure, and we shall consider also a set E ∈ R as a correspondence between R and R .Let us also write D( E ) = { x ∈ R : there exists y ∈ R such that ( x, y ) ∈ E } , which is nothing but a projection of E on the first coordinate. Although the set of Jones functionsis 2 c -lineable (see [21, 25]), it is easy to see that it is not algebrable. Indeed, if f ∈ J ( R ), then f is not even surjective. In [23] the authors defined a dual concept to that of Jones functions but onthe complex plane and they showed that (in the complex setting) this set is actually algebrable.The following result, although of independent interest, provides a relatively easy way to see thatthe set of Jones functions is c -lineable (see also [21, Theorem 2.2] for the best possible result in thisdirection). Proposition 4.1.
Let f ∈ J ( R ) and ϕ ∈ C ( R ) surjective. Then ϕ ◦ f ∈ J ( R ) .Proof. Let Φ : R → R defined by Φ( z, w ) = ( z, ϕ ( w )). Obviously Φ is a continuous function. Let K ⊂ R a closed set with D( K ) uncountable. The set Φ − ( K ) is clearly closed. Besides, we haveD((Φ − ( K ))) ⊃ D( K ). Indeed, if z ∈ D( K ), there exists w such that ( z, w ) ∈ K . As ϕ is surjective,there exists w ′ such that ϕ ( w ′ ) = w . So, Φ( z, w ′ ) = ( z, ϕ ( w ′ )) ∈ K . That is, ( z, w ′ ) ∈ Φ − ( K ), andso z ∈ D((Φ − ( K ))). Therefore, D((Φ − ( K ))) is uncountable, and hence, since f ∈ J ( R ), we have f ∩ Φ − ( K ) = ∅ . Let ( z, w ) ∈ f ∩ Φ − ( K ). Then we have w = f ( z ) and ( z, ϕ ( w )) = Φ( z, w ) ∈ K .That means that ( z, ( ϕ ◦ f )( z )) ∈ K , that is, ( ϕ ◦ f ) ∩ K = ∅ . (cid:3) As we said right before the previous result, and since dim( C ( R )) = c , it suffices to take f ∈ J ( R )and the linear space { ϕ ◦ f : ϕ ∈ C ( R ) } to achieve the c -lineability of J ( R ).The following result shall be useful in the proof of the main result of this section. Proposition 4.2.
Let f ∈ ES ( R ) and ϕ ∈ C ( R ) . Then ϕ ◦ f is a Darboux function.Proof. From Proposition 4.1 the result immediately follows if ϕ is surjective. In general, take I anyconnected subset of R , that is, an interval. Take any f ∈ ES ( R ), then (by definition) f ( I ) = R .Then, ϕ ( f ( I )) is an interval and, therefore, connected. (cid:3) LGEBRABILITY, NON-LINEAR PROPERTIES, AND SPECIAL FUNCTIONS 9
It can be seen (see [4, 24, 25]) that if p is a non-constant polynomial in R and if f ∈ ES ( R ), then p ◦ f is nowhere continuous. This last remark, together with the following corollary to Proposition4.2, shall be useful in the main result of this section. Corollary 4.3.
Let f ∈ ES ( R ) and p any non-constant polynomial in R . Then p ◦ f is nowherecontinuous and Darboux.Remark . One could think that by means of Corollary 4.3 we could easily obtain algebrabilityfrom the fact that the ring of polynomials in R is itself an algebra. Unfortunately this issue is morecomplicated than it might seem at first sight. The reason is that the ring of polynomials in R doesnot contain any infinitely generated algebra. Indeed, in K [ x ] ( K = R or C ), suppose that S is asubalgebra that contains K . Let P be a non-constant polynomial in S and let T be the subalgebraof S generated by 1 and P . Then x is algebraic over T , so K [ x ] is a finitely generated T -module.Note that T is Noetherian and S is a T -submodule of K [ x ], so S is a finitely generated T -module.Hence S is a finitely generated algebra. Obviously, this algebra has infinite dimension ( ω ) as a linearspace. For this reason we need a totally different argument to tackle this problem of algebrability.The following lemma, which is a slight variation of [21, Lemma 2.1] shall be needed throughoutthis section (its proof follows the same idea as in that of [21, Lemma 2.1], and we spare the detailsof it to the interested reader). Lemma 4.5.
Let B ⊂ R be a Bernstein set. There exists f : R → R such that(i) f | B ∩ K = ∅ for every closed set K ⊂ R such that D( K ) is uncountable, and(ii) f ( x ) = 0 for every x / ∈ B . A classical result states that R can be decomposed as the disjoint union of “ c many” Bernsteinsets. Thus, consider such a decomposition R = S i ∈ R B i , where B i is a Bernstein set for every i ∈ R and the family { B i : i ∈ R } is disjoint. Now, use the previous lemma to obtain, for every i ∈ R , afunction f i : R → R such that f i | B i ∩ K = ∅ for every closed set K ⊂ R with D( K ) uncountable.Assume, also, that f i = 0 outside B i . Notice that (by construction) f i · f j = 0 if i = j. (1)Now, let B be the algebra generated by { f i : i ∈ R } , that is, by (1), the linear span of the set { f pi : i ∈ R , p ∈ N } .At this point it is worth mentioning that, by considering the elements in the algebra B , wehave lost the surjectivity (and, thus, the everywhere surjectivity and the properties enjoyed byJones functions) since, for instance, the square of any of the f i ’s is not even surjective. On theother hand, taking into account that J ( R ) ⊂ ES ( R ) and Corollary 4.3 together with that (byconstruction) the f i ’s have disjoint supports it can be seen that every non-zero element of B isnowhere continuous and maps connected sets to connected sets (actually, to unbounded intervals!).Clearly, the family { f i : i ∈ R } is a generator system of B . Let us now see that this generatorsystem is actually minimal. Suppose otherwise, then there would exist f i j , j ∈ { , , , . . . , n } (with i j = i k if j = k ) and a polynomial P ( x , x , . . . , x n ) such that f i = P ( f i , f i , . . . , f i n ) . By (1), there exist polynomials p ( x ), p ( x ), . . . , p n ( x ), such that f i = P ( f i , f i , . . . , f i n ) = p ( f i ) + p ( f i ) + · · · + p n ( f i n ) . But, for every j = 1 , , . . . , n , we have that (by construction) p j ( f i j ) is a constant function on B i .This is a contradiction, since f i is surjective on B i . Thus, we have proved the following result: Theorem 4.6.
The set of nowhere continuous Darboux functions is c -algebrable. Acknowledgements.
The authors wish to thank Prof. Dr. J. L. G´amez-Merino for fruitfulconversations and comments on this paper.
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E-mail address : [email protected] Institute of Mathematics,Technical University of L´od´z,W´olcza´nska 215,93-005 L´od´z, Poland
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