Definition:Pointed 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 pointed extension of $\struct {\R, \tau_d}$:
 * $\tau' = \tau_d \cup \set {H: \exists U \in \tau_d: H = \set x \cup U \cap \Bbb I: x \in U}$

$\tau'$ is then referred to as the pointed irrational extension of $\R$.

Also see

 * Definition:Pointed Rational Extension of Reals


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


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


 * Pointed Extension of Reals is Topology