Characterization of Euclidean Borel Sigma-Algebra

Theorem
Let $\OO^n$, $\CC^n$ and $\KK^n$ be the collections of open, closed and compact subsets of the Euclidean space $\struct {\R^n, \tau}$, respectively.

Let $\JJ_{ho}^n$ be the collection of half-open rectangles in $\R^n$.

Let $\JJ^n_{ho, \text {rat} }$ be the collection of half-open rectangles in $\R^n$ with rational endpoints.

Then the Borel $\sigma$-algebra $\map \BB {\R^n}$ satisfies:


 * $\map \BB {\R^n} = \map \sigma {\OO^n} = \map \sigma {\CC^n} = \map \sigma {\KK^n} = \map \sigma {\JJ_{ho}^n} = \map \sigma {\JJ^n_{ho, \text {rat} } }$

where $\sigma$ denotes generated $\sigma$-algebra.

Proof
By definition of Borel $\sigma$-algebra, $\map \BB {\R^n} = \map \sigma {\OO^n}$.

The rest of the proof will be split in proving the following equalities:


 * $(1): \quad \map \sigma {\OO^n} = \map \sigma {\CC^n}$
 * $(2): \quad \map \sigma {\CC^n} = \map \sigma {\KK^n}$
 * $(3): \quad \map \sigma {\OO^n} = \map \sigma {\JJ_{ho}^n}$
 * $(4): \quad \map \sigma {\JJ_{ho}^n} = \map \sigma {\JJ^n_{ho, \text {rat} } }$