The strange aspect of most compacta

\footnotesizeTo Solomon Marcus, the best of my teachers, for his Real Functions with wonderful features

Abstract

In this paper we describe a compact set K in \R which is typical from the point of view of Baire categories, as it appears when seen from some point x of \R. It matters whether x belongs to K or not! If x¬in 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).