Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?
 A: Various questions about partitioning a topological space $X$ into homeomorphic copies of a topological space $Y$ are discussed by Paul Bankston and Richard J. McGovern, Topological partitions, General Topology and its Applications 10 (1979), 215–229 (pdf). In particular, their Theorem 1.14 says that $\mathbb R$ can be partitioned into Cantor sets. The proof consists of first partitioning $\mathbb R$ into countably many Cantor sets and one set which is isomorphic to the space $\mathbb P$ of irrational numbers, and then using the fact that $\mathbb P$ can be partitioned into $\mathfrak c$ Cantor sets, i.e., $\mathbb N^\mathbb N$ can be partitioned into $\mathfrak c$ copies of $\{0,1\}^\mathbb N$.
From the fact that $\mathbb R$ can be partitioned into Cantor sets, it easily follows that every nonempty perfect subset of $\mathbb R$ can be partitioned into $\mathfrak c$ Cantor sets. Bankston and McGovern prove more general theorems under special set-theoretic assumptions. Namely, they use Martin's axiom to prove that every complete separable metric space with no isolated points can be partitioned into Cantor sets, and they use the continuum hypothesis to prove the same for complete metric spaces of cardinality $\mathfrak c$ with no isolated points.
In my answer to the Math.SE question Partitioning a metric space into Cantor sets, I proved in ZFC that a nonempty Polish space with no isolated points can be partitioned into $\mathfrak c$ Cantor sets.
A: Let $f$ be the x-coordinate of Hilbert's space-filling curve,
whose graph is shown here:
(source: osu.edu)
Then the sets $\{f^{-1}(t)\}_{t\in [0,1]}$
form a partition of the interval [0,1] into Cantor sets.
An easy variation of the above construction produces a partition of the reals (take the point-preimages of $F$, where $F:\mathbb R\to \mathbb R$ is the periodic extension given by $F(t):=\lfloor t\rfloor+f(t-\lfloor t\rfloor)$).
