Definition:Pointed Extension of Reals

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

Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$.

Let $\tau'$ be the topology generated from $\tau_d$ by the addition of all sets of the form $U \cap D$ where $U \in \tau_d$.
 * $\tau' = \tau_d \cup \set {H: \exists U \in \tau_d: H = \set x \cup U \cap D: x \in U}$

$\tau$ is then referred to as a pointed extension of $\R$.

It is usual to focus attention on the two specific cases where $D$ is either the set of rational numbers or the set of irrational numbers:

Also see

 * Pointed Extension of Reals is Topology


 * 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