We suppose that we have a set $\!S \subseteq \mathcal{P}(\mathbb{R})$, so that the following properties are verified:
1). $\space \!S \sim \mathbb{R}$
2). if $\space \!X,\!Y \in \!S \space and \space \!X \neq \!Y, \space$ then $\space \!X \cap\!Y= \varnothing$
3). if $\!X \in \!S$, then $\space \!X \sim \mathbb{R}$
I don't event know where to begin