Borel Sets

A Borel set is simply a member of the Borel σ-algebra \mathcal{B}(\mathbb{R}). That short definition hides an enormous, well-behaved collection: because \mathcal{B}(\mathbb{R}) starts from the open sets and is closed under complement and countable unions, every set you can build from intervals by countably many of those moves is Borel.

Every open set is Borel (they are the generators).
Every closed set is Borel — it is the complement of an open set.
Every interval — open, closed, half-open, a ray — is Borel.
Every single point \{a\} is closed, hence Borel; so any countable set, such as \mathbb{Q}, is a countable union of points and therefore Borel.

One more layer up: Fσ and Gδ

Stacking the operations once more names two classic families. An F_{\sigma} set is a countable union of closed sets; a G_{\delta} set is a countable intersection of open sets. Both are Borel by construction, and they capture sets that are neither open nor closed — \mathbb{Q} is F_{\sigma} (a countable union of points), and the irrationals are G_{\delta}.

The rule of thumb: any set you can actually describe is Borel. Producing a set that is not Borel takes the axiom of choice and a non-constructive argument — you can prove such sets exist, but you can never point at one explicitly.

A Borel set is any member of \mathcal{B}(\mathbb{R}); open, closed, intervals, countable sets, F_{\sigma} and G_{\delta} sets are all Borel.
The Borel sets are closed under complement and countable unions and intersections — never leaving \mathcal{B}(\mathbb{R}).
There are continuum-many Borel sets — vastly fewer than the 2^{\mathfrak{c}} subsets of \mathbb{R}, so non-Borel sets exist (the classic one is a Vitali set, just not nameable explicitly).