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 $\BB$ be the set of sets:

$\BB := \set {H: \exists U \in \tau_d: H = U \cap D}$

Let $\tau^*$ be the topology generated from $\tau_d$ by the addition of all sets of $\BB$.

$\tau^* = \tau_d \cup \BB$

$\tau^*$ 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:

Rational

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

Let $\Q$ denote the set of rational numbers.

Let $\tau^*$ be the indiscrete extension of $\struct {\R, \tau_d}$:

$\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap \Q}$

$\tau^*$ is then referred to as the indiscrete rational extension of $\R$.

Irrational

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

• 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

• Results about indiscrete extensions of $\R$ can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $66 \text { - } 67$. Indiscrete Extension of $R$