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Abstract

Let $X_t$ be a totally disconnected subset of ${\mathbb {R}}$ for each $t\in {\mathbb {R}}$. We construct a partition $\{Y_t\mid t\in {\mathbb {R}}\}$ of ${\mathbb {R}}$ into nowhere dense Lebesgue null sets $Y_t$ such that for every $t\in {\mathbb {R}}$ 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 nonhomeomorphic compact subspaces. Furthermore we prove that for every cardinal number $\kappa $ with $2\leq \kappa \leq 2^{\aleph _0}$ the real line (as well as the Baire space ${\mathbb {R}}\setminus {\mathbb {Q}}$) can be partitioned into exactly $\kappa $ homeomorphic Bernstein sets and also into exactly $\kappa $ mutually nonhomeomorphic Bernstein sets. We also investigate partitions of ${\mathbb {R}}$ into Marczewski sets, including the possibility that they are Luzin sets or Sierpiński sets.