In set theory, a branch of Mathematics, a Borel set is a subset of a topological space obtained by transfinitely iterating the operations of complementation, countable union and countable intersection. The notion of a Borel code gives an absolute way of specifying a borel set of a Polish space in terms of the operations required to form it.

Let ${\displaystyle X}$ be a Polish space. Then it has a countable base. Let ${\displaystyle \left\langle {\mathcal {N}}_{i}|i<\omega \right\rangle }$ enumerate that base (that is, ${\displaystyle {\mathcal {N}}_{i}}$ is the ${\displaystyle i^{\mathrm {th} }}$ basic open set). Now:

• Every natural number ${\displaystyle i}$ is a Borel code. Its interpretation is ${\displaystyle {\mathcal {N}}_{i}}$.
• If ${\displaystyle c}$ is an Borel code with interpretation ${\displaystyle A_{c}}$, then the ordered pair ${\displaystyle \left\langle 0,c\right\rangle }$ is also an Borel code, and its interpretation is the complement of ${\displaystyle A_{c}}$, that is, ${\displaystyle X\setminus A_{c}}$.
• If ${\displaystyle {\vec {c}}}$ is a length-ω sequence of Borel codes (that is, if for every natural number n, ${\displaystyle c_{n}}$ is a Borel code, say with interpretation ${\displaystyle A_{c_{n}}}$), then the ordered pair ${\displaystyle \left\langle 1,{\vec {c}}\right\rangle }$ is an Borel code, and its interpretation is ${\displaystyle \bigcup _{n<\omega }A_{c_{n}}}$.

Then a set is Borel if it is the interpretation of some Borel code.

A Borel code can be looked at as a wellfounded ω-tree and consequently can be coded by an element of the Baire space. This gives a way to construct a surjection from the Baire space to the borel subsets of a Polish space, showing that the number of Borel subsets of a Polish space is bounded above by the cardinality of the Baire space.

The set of Borel codes, the relation x∈${\displaystyle B_{c}}$ are all ${\displaystyle \Pi _{1}^{1}}$, and hence by Schoenfield's Absoluteness Theorem is absolute for inner models M of ZF+DC such that x,c ∈ M.

• Borel set
• Infinity-Borel set

• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-00384-7