Abstract
In this paper, we study the properties of a lower porosity of a set in R2. It turns out that the properties of the lower and upper porosity are symmetrical, except that the main tools for testing the lower porosity are not balls but cones. New families of topologies on R2 generated by the lower porosity are defined. Furthermore, by applying the notion of the lower porosity, we introduce the definition of generalized continuity. Using defined topologies, we study properties of this continuity. We show that the properties of topologies generated by the lower and (upper) porosity are symmetrical.
Highlights
It seems obvious that all properties of the lower porosity and lower porouscontinuity presented in the paper can be extended for sets in Rn and functions f : Rn → R for every n ≥ 2
We proved some properties of the lower porosity of subsets of R2
In Theorem 2, we showed that the lower porosity of the complement of a cone at the vertex can be expressed in the term of the angle of the cone
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The porosity of a set, defined in [1], is the notion of smallness more restrictive than nowhere density and meagerness. It can be defined in arbitrary metric space. In the study of these comparison of classes of functions, it was found that it was possible to strengthen results by using the lower porosity instead of the (upper) porosity. The first aim of this paper is to investigate new properties of the lower porosity of subsets of R2. The second aim of our paper is to describe some properties of lower porouscontinuous functions f : R2 → R. We show that lower porouscontinuous functions and (upper) porouscontinuous functions have the same maximal additive class and different maximal multiplicative class. |^( AB, AC )|—the measure of the angle between lines AB and AC; d( A, BC )—the distance between A and line BC
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
