## Real Analysis Exchange

- Real Anal. Exchange
- Volume 33, Number 1 (2007), 41-50.

### 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.

#### Article information

**Source**

Real Anal. Exchange, Volume 33, Number 1 (2007), 41-50.

**Dates**

First available in Project Euclid: 28 April 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1209398377

**Mathematical Reviews number (MathSciNet)**

MR2402862

**Subjects**

Primary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

**Keywords**

density topology system of paths intersection conditions approximately continuous function

#### Citation

Loranty, Anna. The Intersection Conditions for < s >-Density Systems of Paths. Real Anal. Exchange 33 (2007), no. 1, 41--50. https://projecteuclid.org/euclid.rae/1209398377