Geodesic
$\sigma $-porosity is separably determined
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 219-234
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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.
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.
DOI : 10.1007/s10587-013-0015-3
Classification : 03C15, 28A05, 49J50, 54E35, 54E52, 54H05, 58C20
Keywords: elementary submodel; separable reduction; porous set; $\sigma $-porous set 
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Cúth, Marek; Rmoutil, Martin. $\sigma $-porosity is separably determined. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 219-234. doi: 10.1007/s10587-013-0015-3

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