# Borel sigma-algebra generated by

## Definition

The Borel [ilmath]\sigma[/ilmath]-algebra is the [ilmath]\sigma[/ilmath]-algebra generated by the open sets of a topological space, that is[1]: (where [ilmath](X,\mathcal{O})[/ilmath][Note 1] is any topology)

• [ilmath]\mathcal{B}(X,\mathcal{O}):=\sigma(\mathcal{O})[/ilmath] - if the topology on [ilmath]X[/ilmath] is obvious, we may simply write: [ilmath]\mathcal{B}(X)[/ilmath][1]

## Generators

For a topological space [ilmath](X,\mathcal{O})[/ilmath] the following can be shown:

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

### Proof of claims

Claim 1: [ilmath]\sigma(\mathcal{O})=\sigma(\mathcal{C})[/ilmath]

TODO: Be bothered, just use complements