Large Function Algebras with Certain Topological Properties

Abstract EN

Let F be a family of continuous functions defined on a compact interval.

We give a sufficient condition so that F∪{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 within F∪{0} where F⊂RX or F⊂CX.

We prove that the set of perfectly everywhere surjective functions together with the zero function contains a 2c-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 RR with 2c 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.