Definition:Indiscrete Extension of Reals/Irrational

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Definition

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers.


Let $\tau^*$ be the indiscrete extension of $\struct {\R, \tau_d}$:

$\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap \Bbb I}$


$\tau^*$ is then referred to as the indiscrete irrational extension of $\R$.


Also see

  • Results about the indiscrete irrational extension of $\R$ can be found here.


Sources