According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for each $s \in T$ such that:
(i) $X_{\varnothing} = X$ and $X_s =\bigcup \{X_t : t \in \text{succ}(s)\}$ for all $s \in T$, and
(ii) if $\sigma\in [T]$ then the sequence $X_{σ\restriction 0},X_{σ\restriction 1},...$ converges to a point $x_{\sigma} \in X$.
Here $[T]$ is the set of all infinite branches of $T$.
The authors of [2] state that in [1] Sierpinski proved: A (separable metrizable) space $X$ is absolute $F_{\sigma\delta}$ if and only if $X$ has a Sierpinski stratification.
My question concerns replacing condition (ii) with:
(ii') if $\sigma\in [T]$ then $\bigcap_{n<\omega} X_{σ\restriction n}$ is a singleton.
Apparently, if $(X_s)_{s\in T}$ is a Sierpinski stratification of $X$ then (ii') holds. My main question is whether the definition above is equivalent to the definition with (ii') in the place of (ii). Can this be proved simply by refining the stratification so that at level $n$ of the tree, we intersect each $X_s$ with sets from a countable cover of $X$, such that each set in the cover is a closed neighborhood of diameter $<1/n$?
Also I noticed that neither of these definitions will be identical to what Sierpinski describes in [1]. Is there a more recent reference where Sierpinski's theorem and its variants are proved? Surely there have been more proofs since 1924!
[1] Sierpiński, W., Sur une définition topologique des ensembles (F_{\sigma \delta})., Fund. math. 6, 24-29 (1924). ZBL50.0141.03.
[2] Dijkstra, Jan J.; van Mill, Jan, Erdős space and homeomorphism groups of manifolds, Mem. Am. Math. Soc. 979, v, 62 p. (2010). ZBL1204.57041.
