Large Function Algebras with Certain Topological Properties

Artur Bartoszewicz,Szymon Głąb

doi:10.1155/2015/761924

Artur Bartoszewicz, Szymon Głąb

Open Access

https://doi.org/10.1155/2015/761924

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Abstract

LetFbe a family of continuous functions defined on a compact interval. We give a sufficient condition so thatF∪{0}contains a densec-generated free algebra; in other words,Fis denselyc-strongly algebrable. As an application we obtain densec-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⊂RXorF⊂CX. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a2c-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 ofRRwith2cgenerators which is closed in the pointwise topology and, for any functionfin this algebra, there is an open setUsuch thatf-1(U)is a Bernstein set.

Highlights

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 classes for certain families of functions. It was proved in [1] that the maximal additive class for Darboux real functions is the set of all constant functions
The first papers written in this direction were [2,3,4] and [5,6,7]

Summary

Introduction

The algebraic properties of sets of functions have been considered in analysis for many years. If H ⊆ R is a set of cardinality c and linearly independent over the rationals Q, exp ∘ (rF), r ∈ H, are free generators of an algebra contained in F ∪ {0}. Using [14, Theorem 4.9] and a similar argument, one can prove that the set of all functions from C1, whose derivative is not α-Holder (for any α ∈ (0, 1]) at all but finitely many points, is densely c-strongly algebrable.

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