The countable Borel hierarchy and its limit convention

Work in ZF. Let $(X,\tau )$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) and let ${\omega }_1$ have the meaning in The first uncountable ordinal ${\omega }_1:=\mathrm{\aleph }(\omega )$. Define, for $1\le \alpha <{\omega }_1$,

${\mathrm{\Sigma }}_1^0(X)=\tau ,{\mathrm{\Pi }}_{\alpha }^0(X)=\{X\setminus A:A\in {\mathrm{\Sigma }}_{\alpha }^0(X)\},{\mathrm{\Delta }}_{\alpha }^0(X)={\mathrm{\Sigma }}_{\alpha }^0(X)\cap {\mathrm{\Pi }}_{\alpha }^0(X)\mathrm{.}$

For $1<\alpha <{\omega }_1$, set

${\mathrm{\Sigma }}_{\alpha }^0(X)=\{{\bigcup }_{n\in \mathbb{N}}{A}_n:(A_n{)}_{n\in \mathbb{N}}\in {({\bigcup }_{1\le \beta <\alpha }{\mathrm{\Pi }}_{\beta }^0(X))}^{\mathbb{N}}\}\mathrm{.}$

Thus the summands may have different lower positive ranks, at successors as well as at limits. There is no rank-zero class. A countable union here is an actual sequence, with repetitions allowed.

For existence apply Transfinite recursion to the well-order of positive ordinals below ${\omega }_1$, forming the pair $({\mathrm{\Sigma }}_{\alpha }^0,{\mathrm{\Pi }}_{\alpha }^0)$ at each stage. The formulas use only power sets, the set of sequences, Union and complements in the fixed $X$, so each value is a set. On histories not consisting of the required pairs of subfamilies of $\mathcal{P}(X)$, assign the fixed pair $(\mathrm{\varnothing },\mathrm{\varnothing })$; this makes the recursion rule total. Actual histories have the required type by its construction. This defines all classes uniquely without choice.

The Borel sigma-algebra $\mathcal{B}(X)$ is the intersection of all families of subsets of $X$ containing $\tau$ and closed under complements and unions of sequences. This indexing family is nonempty, since it contains $\mathcal{P}(X)$. Intersections preserve each of the stated closure requirements, so it is the least such family. This definition asserts neither hierarchy exhaustion in ZF nor fixed-rank monotonicity in arbitrary spaces. Empty sets and $X$ occur in every class: they are open and closed at rank one, and constant sequences of them supply all subsequent ranks.