I have the following question:  $\mathbb R^3\setminus \mathbb Q^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski.  someone can give me a help on how to proceed?
Does there exist a set L of pairwise disjoint lines, each line in an axis direction, in $\mathbb{R}^3$ such that their union is equal equal to $\{ x : x \text{ has at least one irrational coordinate}\}$?
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