Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 219-234, 2013

# $\sigma$-porosity is separably determined

## Marek Cúth, Martin Rmoutil

Marek Cúth, Martin Rmoutil, Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8 Karlin, Czech Republic, e-mail: cuthm5am@karlin.mff.cuni.cz, caj@rmail.cz

Abstract: 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.

Keywords: elementary submodel, separable reduction, porous set, $\sigma$-porous set

Classification (MSC 2010): 28A05, 54E35, 58C20

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