Products of non-sigma-lower porous sets

TL;DR

This paper investigates the behavior of lower porosity in Cartesian products of sets within metric spaces, showing that unlike sigma-porosity, lower porosity does not always preserve under product operations, with specific counterexamples and conditions.

Contribution

It constructs explicit counterexamples of non-sigma-lower porous sets whose product is lower porous and establishes conditions under which non-sigma-lower porosity is preserved in products.

Findings

Counterexamples of non-sigma-lower porous sets with lower porous product

Conditions ensuring product of non-sigma-lower porous sets remains non-sigma-lower porous

Characterization of Suslin non-sigma-lower porous sets in complete metric spaces

Abstract

It has been known that the Cartesian product of two Suslin non-sigma-porous sets in topologically complete metric spaces is non-sigma-porous in the product space. The main aim of the present paper is to answer the natural question whether a similar result can be proved for lower porosity. It apears the answer is no, as we construct closed non-sigma-lower porous subsets of the real line A and B such that the Cartesian product of A and B is lower porous. In the present article we provide an example of two closed non-sigma-lower porous sets with lower porous product. On the other hand, we prove the following: Let X and Y be topologically complete metric spaces, let A be a non-sigma-lower porous Suslin subset of X and let B be a non-sigma-porous Suslin subset of Y. Then the product of A and B is non-sigma-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-sigma-lower porous sets in topologically complete metric spaces.