Article
MSC:
03C15,
28A05,
49J50,
54E35,
54E52,
54H05,
58C20 | MR 3035508 | Zbl 1274.54093 | DOI: 10.1007/s10587-013-0015-3
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Keywords:
elementary submodel; separable reduction; porous set; $\sigma $-porous set
elementary submodel; separable reduction; porous set; $\sigma $-porous set
Summary:
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.
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