# Borel σ-algebra generated by intervals on a real line

CA watch $$B _ { mathbb R} = σ (A)$$, when A is:

(a) {$$(a, b): a, b ∈ R$$}

(B) {$$(- ∞, a): a ∈ R$$}

(B) {$$(-, a]: a R$$}

(re) {$$(a, ∞): a ∈ R$$}.

The proof must not be strict.

My solution:

For (a) I discovered that the interval $$(a, b)$$ can be written as:

$$(a, b) = bigcup_ {n = 1} ^ infty (a, b- frac {1} {n})$$ or $$(a, b) = bigcap_ {n = 1} ^ infty (a, b- frac {1} {n}).$$

But, however, I do not know how to show that $$B _ { mathbb R} = σ ($${(a, b): a, b ∈ R}$$)$$.