The strange aspect of most compacta To Solomon Marcus, the best of my teachers, for his Real Functions with wonderful features

摘要

In this paper we describe a compact set K in R~d which is typical from the point of view of Baire categories, as it appears when seen from some point x of R~d. It matters whether x belongs to K or not! If x is not an element of K, then K looks porous (this is easily seen). If x ∈ K, then K looks only σ-porous, but dense at least in a half-sphere (of directions from x). If x is a typical point of K, then K looks even dense (in the whole sphere).