Continuous Functions in Hashimoto Topologies and Their Algebraic Properties

In the paper we consider the Hashimoto topologies on the interval [0, 1] as well as on R, which are connected with the natural topology on R and with some important and well known s-ideals in P( R). We study the families of continuous functions f : [0, 1] -> R with respect to the same Hashimoto topology H(I) (connected with the sigma-ideal I) on the domain and on the range of the considered functions. We show that inside common parts and differences of some such families we can find large (c-generated) free algebras. Some of constructed algebras appear dense in the algebra of the functions which are continuous in the usual sense.