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Let $X$ be a metric space. Then the family of Borel Sets in $X$ is the $\sigma$-algebra generated by the family of open sets.

So if I am not mistaken are we saying that, consider $X$ to be any metric space and the set of open sets of $X$ is a subset of the power set of $X$. Then any set belonging to the $\sigma$-algebra generated by the collection of open sets is called a Borel set. Is this correct or have I misunderstood the definition?

user1314
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  • Yes. You have ${ \mbox{open sets} } \subseteq { \mbox{Borel sets} } \subseteq \mathcal{P}(X)$. – Crostul Apr 16 '15 at 07:37
  • Thanks , could you give any simple examples of this please? Only one I can think of is $\mathbb{Q}$ since it is a countable union of points which are open sets. But I am unable to think up more and I think its maybe because I dont quite understand it intuitively – user1314 Apr 16 '15 at 07:41
  • Borel sets are not intuitive... Look at http://math.stackexchange.com/questions/220248/understanding-borel-sets . I hope this can help. – Crostul Apr 16 '15 at 07:50

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You are correct about the definition of Borel sets. They are elements of the sigma-algebra generated by open sets.

Sander Heinsalu
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