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 $\BB$ be the set of sets defined as:

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

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

$\tau'$ is 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:

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:

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

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

$\tau'$ is referred to as the pointed 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:

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

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

$\tau'$ is referred to as pointed irrational extension of $\R$.

Also see

• 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

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $68 \text { - } 69$. Pointed Extension of $R$