The strange aspect of most compacta

Abstract

In this paper we describe a compact set $K$ in ${\mathbf{R}}^d$ which is typical from the point of view of Baire categories, as it appears when seen from some point $x$ of ${\mathbf{R}}^d$. It matters whether $x$ belongs to $K$ or not! If $x\notin K$ , then $K$ looks porous (this is easily seen). If $x\in K$, then $K$ looks only $\sigma$-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).