An Oscillation Functions on the Real Line

Abstract

By means of a certain well known family $\mathcal {B}$ of subsets of $\mathbb{R}$ fulfilling two conditions we introduce some topologies on $\mathbb{R}$ (in Section 2 we consider the density topology). We observe that the family of the sets $\Omega _f (y):= \left\{ x \in \mathbb{R};\; \; \omega _f(x) \geq y \right\}$ for an arbitrary bounded function $f\:\mathbb{R} \to \mathbb{R}$ (where $\omega _f(x)$ is a kind of ${\cal B}$-oscillation of $f$) has three properties. Then we show that for each family $\left\{ \Omega (y) \right\} _{y \in [0,1]} \subset 2 ^{\mathbb{R}}$ having similar properties and in addition fulfilling conditions $M _1$ and $\mathcal {U}^\prime $ (known from the literature) there is a function $f: \mathbb{R} \to [0,1]$ such that $\Omega _f(y) = \Omega (y)$ for each $y \in[0,1]$. In Section 2 we prove some analogous result for the density topology.