Ideal-like properties of generalized microscopic sets

TL;DR

This paper investigates the properties of generalized microscopic sets, revealing that not all such families form ideals and that their additivity properties vary.

Contribution

It demonstrates that some families of generalized microscopic sets lack ideal structure and explores their weaker additivity properties.

Findings

Not all generalized microscopic set families form an ideal.

Some families have weaker additivity properties.

Some families lack even these weaker properties.

Abstract

We show that not every family of generalized microscopic sets forms an ideal. Moreover, we prove that some of these families have some weaker additivity properties and some of them do not have even that.