TITLE: Difference between Measurable and Borel Measurable function
QUESTION [19 upvotes]: Definition of measurable function: 
If $X$ is measurable space, $Y$ is topological space, then $f:X\to Y$ is measurable provided that $f^{-1}(V)$ is measurable set in $X$ for every open set $V$ in $Y$.
Definition of Borel measurable function: If $f:X\to Y$ is continuous mapping of $X$, where $Y$ is any topological space, $ (X,\mathfrak B)$ is measurable space and $f^{-1}(V)\in\mathfrak B$ for every open set $V$ in $Y$, then $f$ is Borel measurable function.
Both functions are mapping from measurable space to topological space what's the difference between the two definition?

REPLY [13 votes]: The difference is in the $\sigma$-algebra that is part of the definition of measurable space.  In your first definition you say that $X$ is measurable but you don't specify the $\sigma$-algebra; in the second you specify that the $\sigma$-algebra is the collection of all Borel sets.
The $\sigma$-algebra is a collection of sets closed under countably-infinite unions (that's rather rough and ready, but if you need to know more you're better off looking up a definition and some examples).