Unions and ideals of locally strongly porous sets


Abstract: For subsets of $\mathbb R^+ = [0, ∞)$ 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 contained 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 $0$ set is porous at $0$ if and only if $A$ is lower porous at $0$.

Keywords: Local upper porosity, local lower porosity, locally strongly porous set, union of locally porous sets, maximal ideal of locally porous sets

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