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We call Borel set an element of the Borel $\sigma - algebra$ which is defined as the $\sigma - algebra$ generated by the topology of a given topological space. A consequence of this definition is that both open and closed sets are Borel sets.

A Borel set can be either a countable union of closed sets or a countable intersection of open sets. Questions:

  1. How can we say this? Simply because of the definition of $\sigma -algebra$ (closeness with rispect to countable union and countable intersection (using De Morgan's law plus the fact that the complementary of a given set still belong to the $\sigma -algebra$))?

  2. Does it mean that we can see any given Borel set as a countable union of closed sets or countable intersection of open sets?

Any open set $E \subseteq \mathbb{R}^n$ is a countable union of open rectangles (i.e. sets in the form $R = I_1 $ x ... x $I_n$ where $I_j$ are open intervals in $\mathbb{R}$)

  1. why can we say this? Is this related to the previous considerations?

Thank you!

Julian Vené
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  • Regarding #2, the following is a strictly increasing sequence of collections of sets: countable unions along with countable intersections of open sets, countable unions along with countable intersections of sets in the previous collection, countable unions along with countable intersections of sets in the previous collection, etc. Also, there are Borel sets that do not belong to any of these finitely iterated constructions. Indeed, the union of all these collections is itself not closed under countable unions of sets in this union, so the process starts all over again $\ldots$ – Dave L. Renfro Oct 08 '20 at 14:36
  • In general, a $\sigma$ field is closed under countable operations, so this gives you 1. The other, 2, is not true: https://math.stackexchange.com/a/73686/27978 – copper.hat Oct 08 '20 at 15:33

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