Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 205-217, 2013

Products of non-$\sigma$-lower porous sets

Martin Rmoutil

Martin Rmoutil, Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovsk√° 83, 186 75 Praha 8, Czech Republic, e-mail:

Abstract: In the present article we provide an example of two closed non-$\sigma$-lower porous sets $A, B \subseteq\er$ such that the product $A\times B$ is lower porous. On the other hand, we prove the following: Let $X$ and $Y$ be topologically complete metric spaces, let $A\subseteq X$ be a non-$\sigma$-lower porous Suslin set and let $B\subseteq Y$ be a non-$\sigma$-porous Suslin set. Then the product $A\times B$ 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.

Keywords: topologically complete metric space, abstract porosity, $\sigma$-porous set, $\sigma$-lower porous set, Cartesian product

Classification (MSC 2010): 28A05, 54B10, 54E35, 54G20

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