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 $\BB$ be the set of sets defined as:
 * $\BB = \set {\set x \cup \paren {U \cap \Bbb I}: x \in U \in \tau_d}$

Let $\tau'$ be the topology generated from $\BB$.

$\tau'$ is referred to as 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