Products of non-sigma-lower porous sets

Details Products of non-sigma-lower porous sets Products of non-sigma-lower porous sets

2011-12-16

arxiv.org/abs/1112.4378v1

Mathematics

Functional Analysis

Scientific paper

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.

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