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What is this set? Unknown baire generic set

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I'm interested in the set given in this question. For clarity, set

$$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$

where $\\{q_j\\}$ is an enumeration of the rationals. Consider

$$S = \cap_{k = 1}^{\infty} (S_{1/k} \backslash \\{q_k\\}) $$

Since each $S_{1/k} \backslash \\{q_k\\} $ is open and dense, the Baire category theorem gives that $S$ must be baire generic in $\mathbb{R}$, and hence dense. However, $S$ cannot contain any rational number - so what is this set? Moreover, does it depend on the enumeration of $\mathbb{Q}$?

I have some feeling that this is related to Liouville numbers, but lack any references.


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