Unions and ideals of locally strongly porous sets

TL;DR

This paper introduces the concept of coherently porous sets in ^+ and characterizes when unions of strongly porous sets remain strongly porous, revealing structural properties of these sets and their ideals.

Contribution

It defines coherently porous sets and establishes a criterion for the union of strongly porous sets to be strongly porous, advancing the understanding of porosity at zero.

Findings

Union of two strongly porous sets is strongly porous iff they are coherently porous.

Characterizes the intersection of maximal ideals containing strongly porous sets.

Union of a set with a strongly porous set is porous iff the set is lower porous.

Abstract

For subsets of $\mathbb R^+ = [0,\infty)$ we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at $0$ sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals containing in the family of strongly porous at $0$ subsets of $\mathbb R^+$. It is also shown that the union of a set $A \subseteq \mathbb R^+$ with arbitrary strongly porous at $0$ subset of $\mathbb R^+$ is porous at $0$ if and only if $A$ is lower porous at $0$.