Large Function Algebras with Certain Topological Properties

Let F be a family of continuous functions defined on a compact interval. We give a sufficient condition so thatF ∪ {0} contains a dense c-generated free algebra; in other words, F is densely c-strongly algebrable. As an application we obtain dense c-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 withinF∪ {0}whereF⊂ RX orF⊂ C. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a 2-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 R with 2 generators which is closed in the pointwise topology and, for any function f in this algebra, there is an open set U such that f−1(U) is a Bernstein set.


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 ∪{0}) contains a large, nice substructure inside.The first papers written in this direction were [2][3][4] and then [5][6][7].In these papers, the notions contained in the following definition can be found.Definition 1.Let  be a cardinal number.
(1) Let L be a vector space and let  ⊂ L. We say that  is -lineable if ∪{0} contains a -dimensional vector space.
(2) Let L be a Banach space and  ⊂ L. We say that  is spaceable if  ∪ {0} contains an infinite dimensional closed vector space.
(3) Let L be a commutative algebra and let  ⊂ L.
We say that  is -algebrable if  ∪ {0} contains a generated algebra B (i.e., the minimal cardinality of the set generating B equals ).
Bartoszewicz and Głąb in [8] introduced the notion of strong algebrability.Definition 2. Let  be a cardinal number, let L be a commutative algebra, and let  ⊂ L. One says that  is strongly -algebrable if  ∪ {0} 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 2 Journal of Function Spaces topology in R R or C C .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 NH of all nowhere Hölder functions is densely algebrable in [0, 1].In [11], Bastin et al. proved that the set of all nowhere Gevrey functions is densely c-algebrable in The aim of our paper is to formulate, prove, and apply some techniques of constructing dense c-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.

Dense Strong c-Algebrability in 𝐶[0, 1]
It is a simple observation that the set {  → exp() :  ∈ R} is linearly independent in R R .Moreover, if  ⊂ R is linearly independent over Q, then {  → exp() :  ∈ } 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 c-algebrability of the set ([0, 1]) of continuous functions with dense sets of local extrema.Recently, this idea has been further developed in [13,14].
Theorem 3 (see [13]).Let F ⊆ R [0,1] and assume that there exists a function  ∈ F such that  ∘  ∈ F \ {0} for every exponential-like function  : R → R. Then F is strongly c-algebrable.More exactly, if  ⊆ R is a set of cardinality c and linearly independent over the rationals Q, then exp ∘ (),  ∈ , are free generators of an algebra contained in F ∪ {0}.
Using Stone-Weierstrass theorem, it is not difficult to observe that the algebra described in Theorem 3 is dense in [0, 1] 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 F stand for the set of all continuous functions which are differentiable  − 1 times but not differentiable  times at any point of their domains.Let  be the (−1)th antiderivative of a strictly positive nowhere differentiable function.Then by [14,Theorem 4.5], the family F is densely c-strongly algebrable.In turn, using [14, Theorem 4.9] and a similar argument, one can prove that the set of all functions from C 1 , whose derivative is not -Hölder (for any  ∈ (0, 1]) at all but finitely many points, is densely c-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 F does not contain any monotonic function.

Nowhere Constant Continuous Functions. Let 𝐹 :
[, ] → R be a continuous function.Then  is called left nondecreasing at  ∈ (, ] if there is  > 0 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 .
Let  be such that   () is a local minimizer of  with   () >  − .Then for any  ∈ (  (), ) we have (  ()) ≤ () ≤ () which contradicts the definition of  +1 ().In the same manner, we show that  is not left nonincreasing.Therefore  < , since  is left monotonous at .
Proof.Let  =  0 <  1 <  2 < ⋅ ⋅ ⋅ <   =  be any partition of [, ] with the mesh smaller than .We will find a new partition ) contains at most one   and each V  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 | [  −/3,  ] attains its minimum at some   ∈ [  − /3,   ] and maximum at some    ∈ [  − /3,   ].If one of the points   ,    is in (  − /3,   ), then it is a point of local monotonicity and we put it to the new partition.However, it may happen that {  ,    } = {  − /3,   }; that is,   and    are the endpoints of the interval [  − ,   ].We may assume that   =   − /3 and (  ) < (  ).Take any  ∈ (  − /3,   ).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   − /3 and   , and we put it to the new partition.Similarly one can find an appropriate both-sided monotonicity point in (  ,   + /3) which we put into the new partition.
In the next step we will find a refinement for which (i) holds true.To find such a refinement, for every  < , we use Lemma 5 for the restriction  = | [V  ,V +1 ] ,  = V  , and  = V +1 .
From now on we assume that F is flexible.

2. 3 . 6 )
Applications.(1) We say that a continuous function  : [, ] → R is nowhere Hölder, provided that for any  ∈ [, ] and any  ∈ (0, 1] lim sup  →       () −  ()          −       = ∞.(Let us denote the set of all nowhere Hölder functions by NH.It was proved in [14] that  ∘  ∈ NH for any nonconstant analytic function  : R → R and any  ∈ NH.It can be easily seen that if  : [,] → R and   : [, ] → R are nowhere Hölder with () =   (), then  ∪   : [, ] → R is also nowhere Hölder.Therefore, NH is closed under taking continuous piecewise exponential-like transformations.Clearly NH 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  ∈ NH ∩ [0, 1].We may assume that 6 does not hold for .The problem is that  is constant on [−1, 0].