it follows that $$\label{E:2.1} {\mathcal F}'(A)=\big({\mathcal F}(A)\big)'.$$ Indeed $$\begin{aligned}
B\in {\mathcal F}'(A)&\iff  B>A \text{ and } A\cup B\in{\mathcal F}'\\
&\iff  B>A \text{ and } \exists (C_n)\in {\mathcal F}{\!\setminus\!}\{ A\cup B\} \quad C_n\to_{n\to\infty} A\cup B\\
&\iff B>A \text{ and } \exists (B_n)\in {\mathcal F}(A) {\!\setminus\!}\{ B\} \,\,B_n\to_{n\to\infty}  B\iff B\in\big({\mathcal F}(A)\big)'\\
&[\text{Choose $B_n=C_n\setminus A$, for $n$ large enough}].\end{aligned}$$

By transfinite induction we define for each ordinal $\alpha$ the [*$\alpha$-th derivative of ${\mathcal F}$*]{}, by $${\mathcal F}^{(0)}={\mathcal F},\, {\mathcal F}^{(\alpha)}= \big({\mathcal F}^{(\gamma)}\big)',\text{ if }\alpha=\gamma+1,
\text{ and }{\mathcal F}^{(\alpha)}=\bigcap_{\gamma<\alpha}{\mathcal F}^{(\gamma)}, \text{ if } \alpha \text{ is a limit ordinal.}$$

It follows that ${\mathcal F}^{(\beta)}\subset {\mathcal F}^{(\alpha)}$ if $\alpha\le \beta$. By transfinite induction generalizes to $$\label{E:2.2} {\mathcal F}^{(\alpha)}(A)=\big({\mathcal F}(A)\big)^{(\alpha)}, 
\text{ for all $A\in[{\mathbb{N}}]^{<\omega}$ and ordinal  $\alpha$.}$$

Assume that ${\mathcal F}\subset [{\mathbb{N}}]^{<\omega}$ is compact. Since ${\mathcal F}$ is countable and since every countable and compact metric space has isolated points, it follows that for some $\alpha<\omega_1$ the $\alpha$-th derivative of ${\mathcal F}$ is empty and we define $${\text{{\rm CB}}}({\mathcal F})=\min\{\alpha : {\mathcal F}^{(\alpha)}=\emptyset\}.$$ ${\text{{\rm CB}}}({\mathcal F})$ is always a successor ordinal. Indeed, if $\alpha<\omega_1$ is a limit ordinal and ${\mathcal F}^{(\gamma)}\not=\emptyset $ for all $\gamma<\alpha$, it follows that ${\mathcal F}^{(\alpha)}=\bigcap{\mathcal F}^{(\gamma)}\not=\emptyset$.

\[D:2.1\] For ${\mathcal F}$,${\mathcal G}\subset 