# Porosity, Γ*N*- and Γ-Null Sets

# Porosity, Γ*N*- and Γ-Null Sets

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with respect to a family of subspaces are Γₙ-null provided *X* admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by *t*ⁿ logⁿ⁻¹ (1/*t*). The first of these results characterizes Asplund spaces: it is shown that a separable space has separable dual if and only if all its porous sets are Γ₁-null. The chapter first describes porous and σ-porous sets as well as a criterion of Γₙ-nullness of porous sets. It then considers the link between directional porosity and Γₙ-nullness. Finally, it tackles the question in which spaces, and for what values of *n*, porous sets are Γₙ-null.

*Keywords:*
porosity, subspace, bump, Asplund space, separable space, separable dual, porous sets

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