The term Borel hierarchy is used to describe a collection of subsets of 
defined inductively as follows: Level one consists of all open
and closed subsets of 
, and upon having defined levels 
, level 
is obtained by taking countable unions and intersections
of the previous level. In particular, level two of the hierarchy consists of the
collections of all Fsigma and Gdelta
sets while subsequent levels are described by way of the rather confusingly-named
collection of sets of the form 
, 
, 
, 
, 
, etc.
The collection of sets across all levels of the Borel hierarchy is the Borel sigma-algebra. As such, the Borel hierarchy is fundamental to the study of measure theory.
More general notions of the Borel hierarchy (and thus of Borel sets, etc.) are introduced and studied as part of various areas of set theory, topology, and mathematical logic.
