Is Borel-field different from $\sigma$-field? My mathematical statistics book denotes $\sigma$-field as following:
Let $\Bbb B$ be the collection of subsets of $\Bbb C$ where $\Bbb C$ denotes sample space which is the collection of all possible events. Then $\Bbb B$ is $\sigma$-field if 
(1) $\emptyset \in \Bbb B$ and $\exists b \in \Bbb B$ s.t. $\emptyset \subset b$
(2) $C \in \Bbb B \Rightarrow C^c\in \Bbb B $ where $C \in \Bbb C$
(3) $\{C_1, C_2, C_3..\} \in  \Bbb B \Rightarrow \cup_{i=1}^{\infty}C_i \in \Bbb B$ where $\{C_1, C_2, C_3..\}$ is countable collection of subsets of $\Bbb C$
Is this field a specific example of Borel Field? or this field is eqaully defined with Borel Field? 

As answered by @bnd, Borel field is a special case of $\sigma$-field. 


Borel field is the smallest $\sigma$-field containing all open sets and due to complementation it contains all closed sets as well. And for openness and closedness we need a topology. Thus Borel $\sigma$-field is a field on a topological space which is smallest $\sigma$-field containing all open sets and closed sets.


