Large Function Algebras with Certain Topological Properties

Abstract

Let be a family of continuous functions defined on a compact interval. We give a sufficient condition so that contains a dense -generated free algebra; in other words, is densely -strongly algebrable. As an application we obtain dense -strong algebrability of families of nowhere Hölder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras within where or . We prove that the set of perfectly everywhere surjective functions together with the zero function contains a -generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra of with generators which is closed in the pointwise topology and, for any function in this algebra, there is an open set such that is a Bernstein set.

1. Introduction

The algebraic properties of sets of functions have been considered in analysis for many years. One direction of such research is finding the so-called maximal (additive, multiplicative, and so on) classes for certain families of functions. For example, it was proved in [1] that the maximal additive class for Darboux real functions is the set of all constant functions. Recently, a new point of looking on the largeness of sets of functions has appeared. One can call a set , contained in some algebraic structure of functions, a big one if (or ) contains a large, nice substructure inside. The first papers written in this direction were [2–4] and then [5–7]. In these papers, the notions contained in the following definition can be found.

Definition 1. Let be a cardinal number.(1)Let be a vector space and let . We say that is -lineable if contains a -dimensional vector space.(2)Let be a Banach space and . We say that is spaceable if contains an infinite dimensional closed vector space.(3)Let be a commutative algebra and let . We say that is -algebrable if contains a -generated algebra (i.e., the minimal cardinality of the set generating equals ).

Bartoszewicz and Głąb in [8] introduced the notion of strong algebrability.

Definition 2. Let be a cardinal number, let be a commutative algebra, and let . One says that is strongly -algebrable if contains a -generated free algebra.

Let us observe that the notion of spaceability is not a fully algebraic property but it has a topological ingredient (we ask about the existence of closed subspace of given Banach space). Ciesielski et al. in [9] asked about the existence of large linear subspaces, closed in the pointwise or uniform convergence topology in or . So, following this way, one can define spaceability in linear topological spaces.

Some authors were interested in searching for a large substructure with some other topological property, namely, dense lineability (or algebrability) of some classes of functions. For example, Bayart and Quarta in [10] proved that the set of all nowhere Hölder functions is densely -algebrable in . In [11], Bastin et al. proved that the set of all nowhere Gevrey functions is densely -algebrable in .

The aim of our paper is to formulate, prove, and apply some techniques of constructing dense -generated free algebras in the space of continuous functions on a compact interval and to consider the possibility of the existence of closed algebras in some sets of real or complex functions.

2. Dense Strong -Algebrability in

It is a simple observation that the set is linearly independent in . Moreover, if is linearly independent over , then is the set of free generators. In [12] the authors, using the composition of a function with some needed properties with such an exponential function, proved the -algebrability of the set of continuous functions with dense sets of local extrema. Recently, this idea has been further developed in [13, 14].

Let us call, after [13], a function exponential-like of rank whenever is given by the formula for some pairwise distinct nonzero numbers and some nonzero numbers . We have the following.

Theorem 3 (see [13]). Let and assume that there exists a function such that for every exponential-like function . Then is strongly -algebrable. More exactly, if is a set of cardinality and linearly independent over the rationals , then , , are free generators of an algebra contained in .

Using Stone-Weierstrass theorem, it is not difficult to observe that the algebra described in Theorem 3 is dense in if and only if the function is continuous and strictly monotonic. This argument is described in the last section of [14]. To illustrate this, consider the following two examples. Let stand for the set of all continuous functions which are differentiable times but not differentiable times at any point of their domains. Let be the th antiderivative of a strictly positive nowhere differentiable function. Then by [14, Theorem 4.5], the family is densely -strongly algebrable. In turn, using [14, Theorem 4.9] and a similar argument, one can prove that the set of all functions from , whose derivative is not -Hölder (for any ) at all but finitely many points, is densely -strongly algebrable.

However, for many classes of functions, the monotonic representative does not exist. Here we propose some method of construction of a dense algebra even if does not contain any monotonic function.

2.1. Nowhere Constant Continuous Functions

Let be a continuous function. Then is called left nondecreasing at if there is such that for any . Analogously we define a left nonincreasing function at and right nondecreasing (nonincreasing) function at . We say that is a point of local monotonicity, provided that is left nondecreasing or left nonincreasing and is right nondecreasing or right nonincreasing; see [15, 16]. Note that if is a point of local minimum (local minimizer) or a point of local maximum (local maximizer) of , then is a point of local monotonicity. We say that is nowhere constant, provided that its restriction to any open interval is not constant.

Fix a function which is nowhere constant and such that and are points of (one-sided) monotonicity of . For , denote by the largest possible such that is between and for every (here by we mean the singleton ). Such a number always exists by the continuity of . Let and inductively for .

Lemma 4. Let . If , then(i);(ii) is a point of local extremum of ;(iii) is a local minimizer of if and only if is a local maximizer of .

Proof. Since is a point of right local monotonicity of , say is right nondecreasing at , then there is such that for every . Let such that . Since is nowhere constant, then . Hence, .
Now, we will show that is a point of right local monotonicity of . Suppose not, then by the definition of , for . Moreover . Let such that for . Then attains its maximum at some . Since is not right nonincreasing at , then . Moreover for any . This contradicts the definition of .
Proceeding inductively we obtain that or . Note that , are local extrema of . Moreover if is a local minimizer of , then is a local maximizer of and vice versa.

Lemma 5. There is such that .

Proof. Suppose that for every . By Lemma 4, the sequence is strictly increasing. Let . If is left nondecreasing at , then there is such that for every . Let be such that is a local minimizer of with . Then for any we have which contradicts the definition of . In the same manner, we show that is not left nonincreasing. Therefore , since is left monotonous at .
Suppose now that is not right nondecreasing at . Let be a minimizer of on . Then and . Let be such that for every . Then fix such that is a local maximizer of and for . This is possible since and is continuous. Therefore, for , which contradicts the definition of . Similarly one can prove that the assumption that is not right nonincreasing at also leads to contradiction. Hence, is both right nondecreasing and right nonincreasing at . This means that is constant on for some positive , which contradicts the fact that is nowhere constant. This shows that for some .

Lemma 6. Let be nowhere constant and for any . Let . Then there is a partition such that(i) is between and for and ;(ii)the mesh of the partition is smaller than .

Proof. Let be any partition of with the mesh smaller than . We will find a new partition of such that each interval contains at most one and each is a point of local monotonicity of . This new partition will also have a mesh smaller than . We construct it in the following way.
If is a point of local monotonicity of , then remains in the new partition. Otherwise, by the fact that is nowhere constant the restriction attains its minimum at some and maximum at some . If one of the points is in , then it is a point of local monotonicity and we put it to the new partition. However, it may happen that ; that is, and are the endpoints of the interval . We may assume that and . Take any . If is a point of local monotonicity of , then we are done. Assume now that is not a point of local monotonicity of . This means that either is not a point of left monotonicity of or it is not a point of right monotonicity of . We may assume that is not a point of left monotonicity of . Then, attains its maximum on on some and is a both-sided monotonicity point of ; is between and , and we put it to the new partition. Similarly one can find an appropriate both-sided monotonicity point in which we put into the new partition.
In the next step we will find a refinement of for which (i) holds true. To find such a refinement, for every , we use Lemma 5 for the restriction , , and .

The assumption that is nowhere constant in Lemma 6 is essential. To see it, consider a function given by Note that . For every partition with the mesh smaller than 1, there is the largest with . Then, and we may assume that . But there is with , which means that the assertion of Lemma 6 does not hold for . The problem is that is constant on .

Lemma 7. Let be a finite set which is linearly independent over . Let be a nowhere constant continuous function with for any . Then, for any , there are and such that(i);(ii) for , ;(iii)the set is linearly independent over .

Proof. By the previous lemma there are such that(i) is between and for and ;(ii).
We can find real numbers such that the set is linearly independent over , , and . Let . Since is between and , is between and . We have In the second step, we can find real numbers such that the set is linearly independent over , , and . Let . Since is between and , then is between and . We have After steps, the construction is complete.

2.2. Main Theorem

Let be a continuous function. We consider the following operation on . Let be a partition of . Let be such that for , is exponential-like and is continuous. We say that is a continuous piecewise exponential-like transformation of .

We say that a family of continuous functions defined on compact intervals is flexible, provided(1) consists of nowhere constant functions;(2)there is with and for ;(3) for every and for any of its continuous piecewise exponential-like transformation .

From now on we assume that is flexible.

Theorem 8. is densely -strongly algebrable in .

Proof. Let be such that and for any . Using Lemma 7 for and , we find a partition of the unit interval and a continuous function such that(i);(ii) for , ;(iii)the set is linearly independent over .In the next step we use Lemma 7 for and , and we find a refinement of the partition and a continuous function such that(i);(ii) for , ;(iii)the set is linearly independent over and so forth.Inductively we define . Let . By the construction, is linearly independent over . We extend to a linearly independent set over of cardinality . We may assume that there is with . By the assumption, . Let be a polynomial in variables without a constant term. Consider a function . Then, restricted to is of the form where , are pairwise distinct and the vectors of integers are pairwise distinct. Therefore, the numbers , , are distinct as well. Thus, the mapping is a continuous exponential-like transformation of on . Since is closed under continuous piecewise exponential-like transformations, .
This shows that the algebra generated by is a free algebra of generators. To see that is dense in , note that the sequence tends to , and therefore separates the points of . Moreover, note that , which means that the closure of contains all constant functions. Using Stone-Weierstrass theorem, we obtain the assertion.

2.3. Applications

(1) We say that a continuous function is nowhere Hölder, provided that for any and any Let us denote the set of all nowhere Hölder functions by . It was proved in [14] that for any nonconstant analytic function and any . It can be easily seen that if and are nowhere Hölder with , then is also nowhere Hölder. Therefore, is closed under taking continuous piecewise exponential-like transformations. Clearly does not contain a function which is constant on some open interval.

Now, we prove that condition (2) in definition of flexibility is fulfilled. Let . We may assume that (otherwise, consider which is also nowhere Hölder). If for , then we are done. Otherwise, find a maximizer of . Then, for . If for , then an affine transformation of fulfills condition (2) in the definition of a flexible family. Otherwise, find a minimizer of . Then, for . Then, an affine transformation of fulfills condition (2) in the definition of a flexible family. This argument will hold also for the next families.

Finally, by Theorem 8, the set of all nowhere Hölder functions in is densely -strongly algebrable.

(2) We say that a continuous function is Bruckner-Garg of rank (shortly ), provided that there exists a countable set with the property that for all the preimage is a union of a Cantor set with at most many isolated points and for all the preimage is a Cantor set. A function is Bruckner-Garg (shortly ), provided it is Bruckner-Garg of rank for some . Bruckner-Garg functions of rank were investigated in [17], where it was shown that is residual in . By [14, Theorem 4.13] we can easily conclude that is flexible and hence it is densely -strongly algebrable.

(3) Let be the set of all continuous functions such that both sets of their proper local minima and maxima are dense in . Using a similar argument to that in [12], one can prove that the set of all functions from is flexible and thereby it is densely -strongly algebrable.

(4) Denote by the set of all functions in which are nowhere monotonic and differentiable in all but finitely many points; see [18]. It can be shown in a standard way that is flexible; thus, it is densely -strongly algebrable.

3. Closed Algebrability

Aron et al. posed the following problem [19, Problem 4.1]: Characterize when there exists a closed infinite dimensional algebra of functions with a particular “strange” property. Among the classes considered by the authors, there was the family of everywhere surjective functions . In the space or , , we consider two natural topologies, namely, the topology of pointwise convergence—the weakest topology in which each projection is continuous—and the topology of uniform convergence. We will show that the -closure of any nontrivial algebra contains a two-valued function (some characteristic function). Moreover, we will give a sufficient condition for the existence of a closed algebra inside of generators.

The following proposition shows that if is a -closed nontrivial algebra, then contains a two-valued function.

Proposition 9. Let be a subalgebra of or . Then for any the characteristic function of is in .

Proof. Let . Let be the characteristic function of . Take any and . Let for . We need to show that . Let . Put Then, is a polynomial without a constant term such that for any . If , then . Since , . If , then for some and . This shows that .

By , we denote the family of all everywhere surjective functions , that is, functions which map any nonempty open subset of onto . This family appeared at first in terms of algebrability in [7]. By , we denote the family of all perfectly everywhere surjective functions , that is, functions which map any perfect subset of onto . It was proved in [20] that is -strongly algebrable. Since , is -strongly algebrable too. Let stand for the family of all nonconstant Darboux functions. Since any nonconstant Darboux function attains many values, we obtain the following.

Corollary 10. does not contain a nontrivial closed algebra. In particular, the set of all everywhere surjective functions is not 1--closed-algebrable.

Proposition 9 says that any -closed algebra contains two-valued functions. The next step is searching for large -closed algebras in those consisting of functions with a finite range. Note that has a finite range} is an algebra of cardinality . However, the following shows that it does not contain a large -closed (even -closed) algebra.

Theorem 11. Let be an algebra consisting of functions with finite ranges. Then(i)if is finitely generated, then is -closed;(ii)if is not finitely generated, then is not -closed (in particular, it is not -closed).

Proof. (i) Assume that is generated by . Since each has a finite range, we can write where are distinct and is a partition of . Let stand for all finite Boolean combinations of . Clearly, any member of is -measurable. Let be a nonempty atom of the algebra . Then, there are such that . For any , there is a polynomial such that and for . Then, Since is constant on and has finitely many values, there is a polynomial such that is a characteristic function of . Therefore, any -measurable function is in . Since is a -algebra of sets, the family of all -measurable functions is -closed (a pointwise limit of -measurable functions is -measurable).
(ii) Assume now that is not finitely generated. There are which are algebraically independent. As before, and let stand for the set of all finite Boolean combinations of . Suppose that is finite. Again, any characteristic function of an atom in is an algebraic combination of finitely many ’s. Therefore, there is such that any -measurable function is an algebraic combination of . This yields a contradiction. Therefore, is infinite. Hence, we can find pairwise disjoint sets . Define . Since , each is in . Clearly, tends uniformly to .

By , denote the family of all functions which are everywhere discontinuous and is finite. It was proved in [21] that is -algebrable. Immediately we obtain the following.

Corollary 12. does not contain an infinitely generated -closed algebra.

By Proposition 9 and Theorem 11, any infinitely generated -closed algebra contains finite valued and countably valued functions. It turns out that there are large -closed algebras of countably valued functions. Such construction, using the existence of large -independent family, will be used in the next theorem.

A family of subsets of is called -independent, if for every countable set and every where and . By the Tarski theorem [22] there exists a -independent family on of cardinality .

Theorem 13. There is a linear algebra of generators such that for any function there is open set such that is a Bernstein set. In particular, if is the family of all nonmeasurable functions (having no Baire property, nonmeasurable in the sense of Marczewski), then contains a -closed algebra of generators.

Proof. We use the method of independent Bernstein sets which was introduced in [21]. Let be a partition of into many pairwise disjoint Bernstein sets. Let be a -independent family on . For any , put . Let be the -algebra generated by .
Let be the linear algebra generated by . Then each function in is a simple function of the form , where are Boolean combinations of for some distinct . If , then there are which tend pointwisely to . Let be the smallest set such that each is measurable with respect to -algebra generated by . Clearly is countable. There is which does not belong to any , . Consequently, . Therefore, and . Since is not the zero function, for some . There is such that is disjoint with . Since is -measurable, contains a Bernstein set of the form for some . Finally, a set which contains a Bernstein set and is disjoint with some other Bernstein sets is also a Bernstein set.

Let (or ). Fix the partition of (or ). By we define the set Let and let be a polynomial in variables. Let be such that . Then, where . Therefore, the algebra generated by is of form where is a subalgebra of generated by .

Theorem 14. Assume that is unbounded for every . Then, is -closed.

Proof. Note that is metrizable by the metric . To prove that is -closed, take a sequence in tending with respect to to some function . Fix . If is zero on , then obviously . Otherwise, is nonzero. Then, the sequence eventually consists of nonzero functions. Note that for some nonzero polynomials in one variable. By the assumption is unbounded. Note that the sequence is a Cauchy sequence with respect to for . Since is unbounded, then, for distinct polynomials in one variable without constant term, we have . Therefore, the sequence is eventually constant and equal to some polynomial . Thus, .

Corollary 15. There exists a -closed algebra of cardinality and hence -generated, such that consists of perfectly everywhere surjective functions.

Proof. Let be a decomposition of into many Bernstein sets. For any , let be a free generator such that algebra generated by consists of perfectly everywhere surjective functions; the existence of such a function was proved in [20]. Put . Then, is the desired algebra.

For a sequence , put for some increasing . It was proved in [8] that the set of for which is homeomorphic to the Cantor set is strongly -algebrable and comeager. We complete this result with the following.

Theorem 16. The set of those , for which is homeomorphic to the Cantor set, does not contain any nontrivial closed algebra.

Proof. Let be an algebra such that for any the set of limit points is homeomorphic to the Cantor set. Fix nonzero and let . There is a continuous function such that . Let be a sequence of polynomials, tending uniformly to . It is evident that tends in to some with . Since is not homeomorphic to , the algebra cannot be closed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Szymon Głąb has been supported by the National Science Centre, Poland, Grant no. DEC-2012/07/D/ST1/02087.