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.

References

{{Citation

| last1=Jech

| first1=Thomas

| author1-link=Thomas Jech

| title=Set Theory

| publisher=Springer-Verlag

| location=Berlin, New York

| series=Springer Monographs in Mathematics

| year=2003

}}

{{Citation

| last1=Kanamori

| first1=Akihiro

| authorlink=Akihiro Kanamori

| title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings