Abstract
In the paper we consider the Hashimoto topologies on the interval [0,1] as well as on mathbb {R}, which are connected with the natural topology on mathbb {R} and with some important and well known sigma -ideals in mathcal {P}(mathbb {R}). We study the families of continuous functions f:[0,1]rightarrow mathbb {R} with respect to the same Hashimoto topology mathcal {H}(mathcal {I}) (connected with the sigma -ideal mathcal {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 (mathfrak {c}-generated) free algebras. Some of constructed algebras appear dense in the algebra of the functions which are continuous in the usual sense.
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