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 $$\left\langle1,\vec c\right\rangle$$ 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.