Prove Borel Sets

Question

Proof $[a,b], (a,b], [a,b)$ are borel sets.

I read from book, the definition of Borel sets :

Borel sets is smallest $\sigma$-algebra that contains all open sets.

But I cannot understand it.

Please help me.

Answer 1

$[a,b]=\cap_n (a-\frac 1 n, b+\frac 1 n)$, $(a,b]=\cap_n (a, b+\frac 1 n)$ and $[a,b)=\cap_n (a-\frac 1 n, b)$.

Answer 2

1. $ \{a\}$ and $ \{b\}$ are closed, hence Borel.

2. $[a,b]= (a,b) \cup \{a\} \cup \{b\}$.

3. $[a,b)= (a,b) \cup \{a\} .$

4. $(a,b]$ is your turn !