Porosity, Γ‎N- and Γ‎-Null Sets

Abstract

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 tⁿ 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.