User:Chimpionspeak/Borel code

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.

Formal Definition

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

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

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

Observations

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

See Also

• Borel set
• Infinity-Borel set

References

• 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