Definition:Discrete Extension of Reals/Irrational
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Definition
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
- Results about the discrete irrational extension of $\R$ can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous): Part $\text {II}$: Counterexamples: $71$. Discrete Irrational Extension of $R$