(Note: I asked this question at MSE a week ago and received no answer, so I am now reposting it here.)

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define transfinitely: $K^{(0)} = K$; for each ordinal $\alpha$, let $K^{(\alpha+1)}$ be the set of nonisolated points in $K^{(\alpha)}$ when $K^{(\alpha)}$ is equipped with the subspace topology); for $\alpha$ a limit ordinal, set $K^{(\alpha)}= \bigcap_{\beta<\alpha}K^{(\beta)}$).

We shall say here that $K$ has the property (P) if whenever $K^{(\alpha)}\neq \emptyset$ there is a family $\mathcal{F}$ of pairwise-disjoint clopen subsets of $K$ such that $\bigcup \mathcal{F} = K\setminus K^{(\alpha+1)}$ and $\vert U\cap K^{(\alpha)}\vert =1$ for every $U\in \mathcal{F}$.

It is easy to see, for example, that every ordinal has the property (P) when equipped with its natural order topology.

My question is: does the property (P) have an established name in the literature? Or similarly, is it equivalent to some other previously studied property or family of properties taken together? What if $K$ is compact? (I am actually only interested in the compact case).


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