  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.
