Borel hierarchy

Schematic representation of an excerpt from the Borel hierarchy: arrows indicate the transitions between the set systems, the arrows with white squares indicate subset relations

In mathematics and in particular in descriptive set theory , the Borel hierarchy is a step-by-step division of Borel's σ-algebra into a topological space . It represents a constructive structure of all Borel sets. If a property can be proven over all Borel sets, this is often possible by means of transfinite induction over all levels of the Borel hierarchy.

definition

The following set systems are inductively defined over a topological space ( set of open sets): ${\displaystyle (X,{\mathfrak {T}})}$ ${\displaystyle {\mathfrak {T}}}$

• ${\displaystyle \Sigma _{1}^{0}:={\mathfrak {T}}}$.
• ${\displaystyle \Pi _{\alpha }^{0}:=\left\{S\subseteq X|X\setminus S\in \Sigma _{\alpha }^{0}\right\}}$for each countable ordinal number . ${\displaystyle \alpha \geq 1}$
• ${\displaystyle \Sigma _{\alpha }^{0}:=\textstyle \left\{S\subseteq X|\exists A_{0},A_{1},A_{2},\ldots \in \bigcup _{p<\alpha }\Pi _{p}^{0}:S=\bigcup _{i\in \mathbb {N} }A_{i}\right\}}$for each countable ordinal number . ${\displaystyle \alpha >1}$
• ${\displaystyle \Delta _{\alpha }^{0}:=\Sigma _{\alpha }^{0}\cap \Pi _{\alpha }^{0}}$for each countable ordinal number . ${\displaystyle \alpha \geq 1}$

${\displaystyle \Sigma _{1}^{0}}$thus denotes the open sets, the complements of -sets, denotes the sets that can be represented as a countable union of -sets for , and the sets that lie both in and in . ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{p}^{0}}$ ${\displaystyle p<\alpha }$ ${\displaystyle \Delta _{\alpha }^{0}}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{\alpha }^{0}}$

Graduation and monotony properties

• The sets are closed under countable union and finite cut. ${\displaystyle \Sigma _{\alpha }^{0}}$
• The sets are closed under a countable intersection and finite union. ${\displaystyle \Pi _{o}^{0}}$
• The sets are closed under finite intersection, finite union and complement formation. ${\displaystyle \Delta _{\alpha }^{0}}$
• The sets (as well as the sets and the sets) are closed under continuous archetypes, that is: ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Delta _{\alpha }^{0}}$

For a continuous function between topological spaces and is again a -set ( -set, -set) if a -set ( -set, -set) is. ${\displaystyle f\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f^{-1}(A)}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Delta _{\alpha }^{0}}$ ${\displaystyle A\subseteq Y}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Delta _{\alpha }^{0}}$

• ${\displaystyle \Sigma _{\alpha }^{0}\subseteq \Delta _{\alpha +1}^{0}\supseteq \Pi _{\alpha }^{0}}$for all countable ordinal numbers . ${\displaystyle \alpha }$

In uncountable Polish rooms, which always have the cardinality (see Beth function ), these inclusions are always strict, while in countable Polish rooms all subsets of the room are already contained. ${\displaystyle \beth _{1}}$ ${\displaystyle \Sigma _{2}^{0}}$

Relation to Borel's σ-algebra

The union of all set systems of the Borel hierarchy forms exactly the Borel σ-algebra , i.e. i. the smallest σ-algebra that contains all open sets of topological space.

The fact that every set in the Borel hierarchy must be contained in Borel's σ-algebra follows directly from the final properties of a σ-algebra: If there were sets in the Borel hierarchy that are not contained in Borel’s σ-algebra, there would be one smallest ordinal number , so that one contains such (because the ordinal numbers are well ordered ), which is equivalent to that such contains, since σ-algebras are closed with complement formation. However, this element is the union of countable many elements of which are all Borel sets. Thus the element would also have to be contained in the σ-algebra, since σ-algebras are closed with regard to countable union. ${\displaystyle n}$ ${\displaystyle \Sigma _{n}^{0}}$ ${\displaystyle \Pi _{n}^{0}}$ ${\displaystyle \textstyle \bigcup _{m<n}\Pi _{m}^{0}}$

Conversely, all open sets are contained in the Borel hierarchy and the sets of the Borel hierarchy are closed under complement formation and countable union: The former follows directly from the definition of complements, the latter can be shown as follows: Let countable many in the Borel Given hierarchy occurring quantities . For each there is an ordinal number , so that eventually the sets appear in the hierarchy. The following then applies to the supremum , and the supremum of a set of ordinal numbers is their union, thus is a countable union of countable sets and thus in turn a countable ordinal number. Now it becomes clear why countable ordinal numbers were chosen. ${\displaystyle \Pi _{n}^{0}}$ ${\displaystyle S_{0},S_{1},S_{2},\ldots }$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle \alpha _{i}}$ ${\displaystyle S_{i}\in \Sigma _{\alpha _{i}}^{0}}$ ${\displaystyle {\hat {\alpha }}:=\sup _{i\in \mathbb {N} }\alpha _{i}}$ ${\displaystyle \textstyle \bigcup _{i\in \mathbb {N} }S_{i}\in \Sigma _{\hat {\alpha }}^{0}}$ ${\displaystyle {\hat {\alpha }}}$

Relation to the projective hierarchy

The projective hierarchy is defined for Polish spaces based on the Borel hierarchy , which is based on the analytical sets , the projections of Borel sets. According to Suslin's theorem , the Borel sets in a Polish space are precisely those sets that are analytic and whose complement is also analytic.

Dual definition of closed sets

The Borel hierarchy can also be defined based on the closed sets:

• ${\displaystyle \Pi _{1}^{0}}$ be the set of all closed sets.
• ${\displaystyle \Sigma _{\alpha }^{0}:=\left\{S\subseteq X|X\setminus S\in \Pi _{\alpha }^{0}\right\}}$for each countable ordinal number . ${\displaystyle \alpha \geq 1}$
• ${\displaystyle \textstyle \Pi _{\alpha }^{0}:=\left\{S\subseteq X|\exists A_{0},A_{1},A_{2},\ldots \in \bigcup _{p<\alpha }\Sigma _{\alpha }^{0}:S=\bigcap _{i\in \mathbb {N} }A_{i}\right\}}$for each countable ordinal number . ${\displaystyle \alpha >1}$

The -sets are thus defined as the countable intersection of -sets for . ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Sigma _{p}^{0}}$ ${\displaystyle p<\alpha }$

nomenclature

Felix Hausdorff introduced the following names for the levels of the hierarchy to finite ordinals: , , , , , etc., see also G _δ - and F _σ quantities . The uniform notation , , which indicates the analogy to the arithmetic hierarchy in recursion, was introduced by John Addison 1959th ${\displaystyle F_{\sigma }:=\Sigma _{2}^{0}}$ ${\displaystyle G_{\delta }:=\Pi _{2}^{0}}$ ${\displaystyle F_{\sigma \delta }:=\Pi _{3}^{0}}$ ${\displaystyle G_{\delta \sigma }:=\Sigma _{3}^{0}}$ ${\displaystyle F_{\sigma \delta \sigma }:=\Sigma _{4}^{0}}$ ${\displaystyle G_{\delta \sigma \delta }:=\Pi _{4}^{0}}$ ${\displaystyle \Sigma _{\alpha }^{0}}$ ${\displaystyle \Pi _{\alpha }^{0}}$ ${\displaystyle \Delta _{\alpha }^{0}}$

Individual evidence

1. ↑ Descriptive Set Theory (PDF; 643 kB) , lecture notes by David Marker, 2002
2. ^ Addison, John W .: Separation principles in the hierarchy of classical and effective descriptive set theory , Fundamenta Mathematicae XLVI, pp. 123-135, 1959. pdf .