# Article

Full entry | PDF   (0.2 MB)
Keywords:
topologically complete metric space; abstract porosity; $\sigma$-porous set; $\sigma$-lower porous set; Cartesian product
Summary:
In the present article we provide an example of two closed non-$\sigma$-lower porous sets $A, B \subseteq \mathbb R$ 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.
References:
[1] Engelking, R.: General Topology. Rev. and Compl. Ed., Sigma Series in Pure Mathematics 6. Heldermann Berlin (1989). MR 1039321
[2] Koc, M., Zajíček, L.: On Kantorovich's result on the symmetry of Dini derivatives. Commentat. Math. Univ. Carol. 51 (2010), 619-629. MR 2858265 | Zbl 1224.26021
[3] Zajíček, L.: Porosity and $\sigma$-porosity. Real Anal. Exch. 13 (1987/88), 314-350. MR 0943561 | Zbl 0666.26003
[4] Zajíček, L.: Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces. Czech. Math. J. 41 (1991), 288-296. MR 1105445 | Zbl 0768.58005
[5] Zajíček, L.: Products of non-$\sigma$-porous sets and Foran systems. Atti Semin. Mat. Fis. Univ. Modena 44 (1996), 497-505. MR 1428780 | Zbl 0877.54023
[6] Zajíček, L.: On $\sigma$-porous sets in abstract spaces. Abstr. Appl. Anal. 5 (2005), 509-534. DOI 10.1155/AAA.2005.509 | MR 2201041
[7] Zajíček, L., Zelený, M.: Inscribing closed non-$\sigma$-lower porous sets into Suslin non-$\sigma$-lower porous sets. Abstr. Appl. Anal. 3 (2005), 221-227. DOI 10.1155/AAA.2005.221 | MR 2197116 | Zbl 1091.28001
[8] Zelený, M., Pelant, J.: The structure of the $\sigma$-ideal of $\sigma$-porous sets. Commentat. Math. Univ. Carol. 45 (2004), 37-72. MR 2076859 | Zbl 1101.28001

Partner of