A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open
sets or closed sets by repeatedly taking countable
unions and intersections.
Formally, the class 
of Borel sets in Euclidean space 
is the smallest collection of sets
that includes the open sets and closed
sets such that if 
,
, 
, ... are in 
, then so are 
, 
, and 
, where 
is a set difference (Croft
et al. 1991).
The set of rational numbers is a Borel set, as is the Cantor set.
Orthogonal projections need not preserve Borel sets. Krokhalev (2026) constructed a bounded set in the plane that is Borel but whose orthogonal projection onto every line is non-Borel. These projections are nevertheless analytic sets.
