Using transfinite induction to split $R$ to continuum many pairwise disjoint subsets of $R$

Question

I am looking for different ways to partition $R$. I know some like :

(1) Define a relation as following $$x\sim y \ \text{iff} \ x-y\in\mathbb Q(x,y\in\mathbb R)$$. The equivalence classes have the form $[r]=r+\mathbb Q$ and clear they are countable dense and pairwise disjoint and $\mathbb R=\bigcup_{r\in\mathbb R} [r]$.

(2) Let $P$ be the family of all nonempty perfect subsets of $\mathbb R$ so $|P\times\mathbb R|=c.$ Then we can enumerate $P\times\mathbb R$ as follows $\{<P_{\xi},y_{\xi}>\colon\xi<c\}$. Notice that each perfect will appear $c$ many time as first pair. We will construct by induction on $\xi$ a sequence $\{x_\xi\colon \xi<c\}$ such that

$$x_{\xi}\in P_{\xi}\setminus\{x_{\zeta}\colon \zeta<\xi\}$$

Since each $x_{\lambda}\neq x_{\xi}$ for all $\lambda<\xi$ so we can define $f$ on $\{x_\xi\colon \xi<c\}$ such that $f(x_{\xi})=y_{\xi}$ and $f(x)=0$ otherwise, Thus, $f$ has a desired property. It is not hard to see $f^{-1}(r)$ for each $r\in\mathbb R$ and perfect set $P$ we have $$f^{-1}(r)\cap P\neq\emptyset$$ and $$f^{-1}(r)\cap (R\setminus P)\neq\emptyset$$ $\mathbb R=\bigcup_{r\in\mathbb R} f^{-1}(r).$ Notice that $f^{-1}(r)$ is dense as well. $c$ is the cardinality for $\mathbb R.$

(3) Also, in John C, Oxtoby, Measure and Category, $\mathbb R$ can be written as union of meager set and null set.
My question is I want to see more interesting partition by using transfinite induction. Please Share you your ways if you know some. Thank in advance.

Answer 1

Quite a few fairly strong such partition results are known. A few can be found in the following references, and googling their titles will give you many more: Sur une décomposition d'un intervalle en une infnité non dénombrable d'ensembles non mesurables by Luzin/Sierpiński (1917), Sur la décomposition de l'espace euclidien en ensembles homogènes by Erdős/Marcus (1957; Zbl review), Point Set Theory by John Clifford Morgan (1990; see p. 152-154, pp. 245-248, and the references he gives), A nonmeasurable partition of the reals by Paula Ann Kemp (2001).

Regarding applications of transfinite induction for results such as your (1)-(3), you'll find a huge number by looking page-by-page through the earliest volumes (1920s and 1930s) of the journal Fundamenta Mathematicae.