The Borel hierarchy classifies Borel subsets of a topological space 
by countable operations starting
from its open sets. The first levels are 
, the open sets, and 
, the closed
sets. For every countable ordinal 
, 
consists of countable unions of sets in 
, and 
consists of the complements
of sets in 
. Equivalently, 
consists of countable intersections of sets from
lower 
levels. The class 
is 
.
For example, 
is the class of Fsigma sets, and
is the class of Gdelta
sets. The union of the levels over all countable ordinals is the Borel
sigma-algebra, so the Borel hierarchy is fundamental to 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.
