Countable Set Sigma Algebra is Generated by Singletons

Last updated Nov 1, 2022

Let $S$ be a Set. Then $\sigma({{s} \subset S : s \in S})$ is the Countable Set Sigma Algebra on $S$.

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1. This establishes $\sigma({{x} \subset \mathbb{R} : x \in \mathbb{R}}) \subsetneq \mathcal{B}(\mathbb{R})$ (since $(0,1)$ is Uncountable and $(0, 1)^{C}$ is Uncountable).

• Axiom Schema of Specification
• Axiom of Power Set
• Borel Sigma Algebra