Definition:Indiscrete Extension of Reals/Irrational

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

 * Definition:Indiscrete Rational Extension of Reals


 * Definition:Pointed Extension of Reals:
 * Definition:Pointed Rational Extension of Reals
 * Definition:Pointed Irrational Extension of Reals


 * Definition:Discrete Extension of Reals:
 * Definition:Discrete Rational Extension of Reals
 * Definition:Discrete Irrational Extension of Reals