Comparison of families of S-continuous functions

Abstract

Abstract Some sequences S of measurable sets tending to zero generate density-like topologies T S on the real line finer than the natural topology T n a t . For a topology T S we consider the metric space C S , S of S -continuous functions f : ( R , T S ) → ( R , T S ) , and we consider the metric spaces C n a t , n a t and C + of continuous functions and of right continuous functions, respectively, all spaces endowed with the supremum metric. We prove that if S is right-sided, then the family C + ∩ C S , S is strongly porous in C S , S and dense in C + . We prove that if for some α − 1 the function αx is S -continuous, then S is not one-sided and C n a t , n a t ∩ C S , S is strongly porous in C S , S and dense in C n a t , n a t .