On decompositions of the real line

Let X_t be a totally disconnected subset of the real line R for each t in R. We construct a partition {Y_t | t in R} of R into nowhere dense Lebesgue null sets Y_t such that for every t in R there exists an increasing homeomorphism from X_t onto Y_t. In particular, the real line can be partitioned into 2^{aleph_0} Cantor sets and also into 2^{aleph_0} mutually non-homeomorphic compact subspaces. Furthermore we prove that for every cardinal number k with 2 \leq k \leq 2^{aleph_0} the real line (as well as the Baire space R\Q) can be partitioned into exactly k homeomorphic Bernstein sets and also into exactly k mutually non-homeomorphic Bernstein sets. We also investigate partitions of R into Marczewski sets, including the possibility that they are Luzin sets or Sierpinski sets.