Definition:Indiscrete 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 $\tau^*$ 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
- Results about the indiscrete irrational extension 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: $67$. Indiscrete Irrational Extension of $R$