Polish Spaces and the Borel Hierarchy

Descriptive set theory studies the complexity of subsets of Polish spaces, using hierarchies that classify sets according to the logical operations needed to define them.

Polish Spaces

Definition9.1Polish space

A Polish space is a separable completely metrizable topological space. Examples include $\mathbb{R}$ , $\mathbb{R}^n$ , the Cantor space $2^\omega = \{0,1\}^{\mathbb{N}}$ , the Baire space $\omega^\omega = \mathbb{N}^{\mathbb{N}}$ , separable Banach spaces, and closed subspaces of Polish spaces.

Baire space $\omega^\omega$ with the product topology (of discrete $\omega$ ) is the canonical "universal" Polish space: every Polish space is a continuous image of $\omega^\omega$ .

The Borel Hierarchy

Definition9.2Borel hierarchy

The Borel sets in a Polish space $X$ are generated from open sets by countable unions, countable intersections, and complementation. They are classified into levels:

$\boldsymbol{\Sigma}^0_1$ = open sets.
$\boldsymbol{\Pi}^0_1$ = closed sets (complements of open).
$\boldsymbol{\Sigma}^0_{\alpha+1}$ = countable unions of $\boldsymbol{\Pi}^0_\alpha$ sets.
$\boldsymbol{\Pi}^0_\alpha$ = complements of $\boldsymbol{\Sigma}^0_\alpha$ sets.
$\boldsymbol{\Sigma}^0_\lambda = \boldsymbol{\Pi}^0_\lambda = \bigcup_{\alpha < \lambda} \boldsymbol{\Sigma}^0_\alpha$ for limit $\lambda$ .

The hierarchy stabilizes at $\omega_1$ : $\mathbf{Borel} = \bigcup_{\alpha < \omega_1} \boldsymbol{\Sigma}^0_\alpha$ .

ExampleBorel sets at various levels

$\boldsymbol{\Sigma}^0_1$ : open intervals, open balls.
$\boldsymbol{\Pi}^0_1$ : closed intervals, closed sets.
$\boldsymbol{\Sigma}^0_2$ ( $F_\sigma$ ): countable unions of closed sets ( $\mathbb{Q}$ , for instance).
$\boldsymbol{\Pi}^0_2$ ( $G_\delta$ ): countable intersections of open sets (irrationals $\mathbb{R} \setminus \mathbb{Q}$ ).
$\boldsymbol{\Sigma}^0_3$ ( $G_{\delta\sigma}$ ): countable unions of $G_\delta$ sets.

Each level is strictly contained in the next (in uncountable Polish spaces), so the hierarchy does not collapse.

Borel Measurability

Theorem9.1Properties of Borel sets

Every Borel set in a Polish space is:

Lebesgue measurable (in $\mathbb{R}^n$ ).
Has the Baire property (differs from an open set by a meager set).
Has the perfect set property: every uncountable Borel set contains a perfect subset (and hence has cardinality $\mathfrak{c}$ ).

These are the "regularity properties." They hold for Borel sets in ZFC but can fail for arbitrary sets.

RemarkThe Borel isomorphism theorem

Any two uncountable Polish spaces are Borel isomorphic: there exists a bijection that preserves the Borel $\sigma$ -algebra. Thus $\mathbb{R}$ , $[0,1]$ , $\omega^\omega$ , and $2^\omega$ are all "the same" from the Borel perspective. This remarkable rigidity means that descriptive set theory on one Polish space transfers to all others.