Definition:Pointed 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:
- $\BB = \set {\set x \cup \paren {U \cap \Bbb I}: x \in U \in \tau_d}$
Let $\tau'$ be the topology generated from $\BB$.
$\tau'$ is referred to as pointed irrational extension of $\R$.
Also see
- Results about the pointed irrational 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: $69$. Pointed Irrational Extension of $R$