Definition:Rational Extension in Plane

Definition

Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology.

Let $D$ denote the set:

$D := \set {\tuple {x, y}: x \in \Q, y \in \Q}$

where $\Q$ denotes the set of rational numbers.

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 the rational extension in the plane.

Also see

• Rational Extension in Plane is Topology

• Results about the rational extension in the plane can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $72$. Rational Extension in the Plane