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First off, I want to point out a popular topology for the real line. We know it commonly as the Michael line. It is given by

$\mathfrak{T}_M =\{U \cup I: U$ is open under usual topology, $I \subset \mathbb{R} \setminus \mathbb{Q} \}$.

But is there a name for the topology given by

$\mathfrak{T} =\{U \cup Q: U$ is open under usual topology, $Q \subset \mathbb{Q} \}$?

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This is defined in Counterexamples in Topology as space #70, Discrete rational extension of $\mathbb R$.

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    $\begingroup$ Interestingly we don't have "Michael Line" as an alias of S63; this should be fixed. $\endgroup$ Commented Dec 13, 2022 at 0:03
  • $\begingroup$ Thanks! I appreciate the feedback! $\endgroup$ Commented Dec 14, 2022 at 5:38

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