Intersection properties of typical compact sets

We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. We study the "hitting probabilities" of compact sets in the sense of Baire category. In the end we study the arithmetic properties of typical compact sets in $[0,1]$ and the "hitting probabilities" of continuous functions.