Unions and ideals of locally strongly porous sets

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 $0$.

arXiv comment28 pages

References in corpus (1)

• Porosity of Collet-Eckmann Julia sets