The Intersection Conditions for <s>-Density Systems of Paths

Abstract

We investigate the intersection conditions for an $\s$-density system of paths. We show, for example, that for every unbounded and nondecreasing sequence of positive numbers $\s$ such that $ \liminf_{n\rightarrow \infty }\frac{s_n}{s_{n+1}} =0$, there exists a system of paths connected with $\s$-density points which does not satisfy the intersection conditions. Moreover, we show that a function $f\:mathbb{R}\rightarrow \mathbb{R}$ is $\s$-approximately continuous if and only if $f$ is continuous with respect to some $\s$-density system of paths.