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.

Three pictures, one family

The definition is abstract, so pin it to the line. Play the figure: a closed interval, a singleton that is not even rational, and a spray of rationals. None of these is an open set (the interval has endpoints, a point has no breathing room, the rationals are not open), yet every one of them is Borel — because each is built from open sets by complement and countable union. That is the whole point of generating a σ-algebra: you keep the sets you can describe, not merely the sets that look open.

A worked construction: why a point is Borel

Take the singleton \{a\}. Its complement is the union of two open rays,

\{a\}^{c} = (-\infty, a) \cup (a, \infty),

which is open, hence Borel. The complement of a Borel set is Borel, so \{a\} itself is Borel. Once every singleton is in, any countable set is a countable union of singletons and is Borel too — that is how \mathbb{Q} = \bigcup_{n=1}^{\infty}\{q_n\} gets in, after you enumerate the rationals. The same argument, with the endpoints included rather than excluded, shows that every closed interval [a,b] is Borel: it is the complement of (-\infty,a)\cup(b,\infty).

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 complement of that F_{\sigma}, or equivalently \bigcap_n U_n where each U_n is the open set obtained by punching a tiny interval out around the nth rational).

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).

Not open does not mean not Borel. Closed intervals, singletons, and \mathbb{Q} are all Borel, and none of them is open. Open sets generate the Borel σ-algebra; they are not the only members.
Borel is not about length. Being Borel is a membership question (is the set in \mathcal{B}(\mathbb{R})?). Whether it has length zero is a later, different question — Lebesgue measure will assign \lambda(\mathbb{Q}) = 0, but that does not decide whether \mathbb{Q} is Borel (it is).