On decompositions of the real line
Tom 165 / 2021
Streszczenie
Let 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.