Continuous Functions in Hashimoto Topologies and Their Algebraic Properties

Abstract

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