Definition:Indiscrete 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 $U \in \tau_d$.
 * $\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap D}$

$\tau^*$ is then referred to as an indiscrete 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

 * Indiscrete Extension of Reals is Topology


 * 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