Definition:Discrete 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 defined as:

$\BB = \tau_d \cup \set {\set x: x \in D}$

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

$\tau^*$ is referred to as a discrete 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 $\BB$ be the set of sets defined as:

$\BB = \tau_d \cup \set {\set x: x \in \Q}$

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

$\tau^*$ is referred to as the discrete 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 $\BB$ be the set of sets defined as:

$\BB = \tau_d \cup \set {\set x: x \in \Bbb I}$

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

$\tau^*$ is referred to as the discrete irrational extension of $\R$.

Also see

• Discrete Extension of Reals is Topology

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

• Definition:Pointed Extension of Reals:
  • Definition:Pointed Rational Extension of Reals
  • Definition:Pointed Irrational Extension of Reals

• Results about discrete 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: $70 \text { - } 71$. Discrete Extension of $R$