Previous |  Up |  Next


elementary submodel; separable reduction; porous set; $\sigma $-porous set
We prove a separable reduction theorem for $\sigma $-porosity of Suslin sets. In particular, if $A$ is a Suslin subset in a Banach space $X$, then each separable subspace of $X$ can be enlarged to a separable subspace $V$ such that $A$ is $\sigma $-porous in $X$ if and only if $A\cap V$ is $\sigma $-porous in $V$. Such a result is proved for several types of $\sigma $-porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem of L. Zajíček on differentiability of Lipschitz functions on separable Asplund spaces to the nonseparable setting.
[1] Cúth, M.: Separable reduction theorems by the method of elementary submodels. Fundam. Math. 219 191-222 (2012). DOI 10.4064/fm219-3-1 | MR 3001239
[2] Dow, A.: An introduction to applications of elementary submodels to topology. Topology Proc. 13 (1988), 17-72. MR 1031969 | Zbl 0696.03024
[3] Kruger, A. Y.: On Fréchet subdifferentials. J. Math. Sci., New York 116 (2003), 3325-3358. DOI 10.1023/A:1023673105317 | MR 1995438 | Zbl 1039.49021
[4] Kubiś, W.: Banach spaces with projectional skeletons. J. Math. Anal. Appl. 350 (2009), 758-776. DOI 10.1016/j.jmaa.2008.07.006 | MR 2474810 | Zbl 1166.46008
[5] Kunen, K.: Set Theory. An Introduction to Independence Proofs. 2nd print. Studies in Logic and the Foundations of Mathematics, 102 North-Holland, Amsterdam (1983). MR 0756630 | Zbl 0534.03026
[6] Lindenstrauss, J., Preiss, D., Tišer, J.: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. Annals of Mathematics Studies 179 Princeton, NJ: Princeton University Press (2012). MR 2884141 | Zbl 1241.26001
[7] Rmoutil, M.: Products of non-$\sigma$-lower porous sets. Czech. Math. J. 63 (2013), 205-217. DOI 10.1007/s10587-013-0014-4 | MR 3035507
[8] Zajíek, L.: Sets of $\sigma$-porosity and sets of $\sigma$-porosity $(q)$. Čas. Pěst. Mat. 101 (1976), 350-359. MR 0457731
[9] Zajíek, L.: A generalization of an Ekeland-Lebourg theorem and the differentiability of distance functions. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 3 (1984), 403-410. MR 0744405
[10] Zajíek, L.: Porosity and $\sigma$-porosity. Real Anal. Exch. 13 (1987/88), 314-350. MR 0943561
[11] Zajíek, L.: Fréchet differentiability, strict differentiability and subdifferentiability. Czech. Math. J. 41 (1991), 471-489. MR 1117801
[12] Zajíek, L.: Products of non-$\sigma$-porous sets and Foran systems. Atti Semin. Mat. Fis. Univ. Modena 44 (1996), 497-505. MR 1428780
[13] Zajíek, L.: On $\sigma$-porous sets in abstract spaces. Abstr. Appl. Anal. 5 (2005), 509-534. DOI 10.1155/AAA.2005.509 | MR 2201041
[14] Zajíek, L., Zelený, M.: Inscribing compact non-$\sigma$-porous sets into analytic non-$\sigma$-porous sets. Fundam. Math. 185 (2005), 19-39. MR 2161750
[15] 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
EuDML logo