Borel sigma-algebra generated by

Definition

The Borel $\sigma$-algebra is the $\sigma$-algebra generated by the open sets of a topological space, that is: (where $(X,\mathcal{O})$ is any topology)

• $\mathcal{B}(X,\mathcal{O}):=\sigma(\mathcal{O})$ - if the topology on $X$ is obvious, we may simply write: $\mathcal{B}(X)$

Generators

For a topological space $(X,\mathcal{O})$ the following can be shown:

Claim	Proof route	Comment
$\mathcal{B}(X):=\sigma(\mathcal{O})$	Trivial (by definition)
$\mathcal{B}(X)=\sigma(\mathcal{C})$ - the closed sets	$\mathcal{B}(X):=\sigma(\mathcal{O})=\sigma(\mathcal{C})$ - see claim 1 below

Proof of claims

[<collapsible-expand>]

Claim 1: $\sigma(\mathcal{O})=\sigma(\mathcal{C})$

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TODO: Be bothered, just use complements

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See also

• Borel $\sigma$-algebra - a special case, where $\mathcal{B}:=\mathcal{B}(\mathbb{R},\vert\cdot\vert)$ and $\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n,\vert\cdot\vert)$