Borel's field

I don't believe that your question can be answered the way you want for any realistic case. The simplest case of interest is the Borel field for the real line under the ordinary topology, i.e. topology based on open intervals. The topology (open sets) consists of all unions and all finite intersections of open intervals. A sigma field is a collection of sets closed under countable unions and intersections. The Borel field is the smallest sigma field containing the open sets.

There is no way to illustrate this construction with Venn diagrams.