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? 

A sigma field on a non-emptyset $X$ is a collection $\mathcal{F}\subseteq 2^X$ that contains $\emptyset$, is closed under complementation and is closed under countable unions.
A Borel field is a sigma field $\mathcal{F}$ that is defined on a topological space $(X, \mathcal{T})$ such that $\mathcal{T} \subseteq \mathcal{F}$. Most authors require that $\mathcal{F}$ is generated by the topology $\mathcal{T}$, that is $\mathcal{F}$ is the smallest sigma-algebra on $X$ containing the topology $\mathcal{T}$.
