Thank you, now I understand. All of this started in my head while trying to understand the proof of the following.
I tried to expand the proof of it, using the additional material you provided. From the comment above we can deduce the following
Lemma.
- pot is the (i.e. unique) highest amongst all the sets including inc and excluding exc.
- t\in pot if and only if feerate(t)> feerate(pot).
Proposition. Let B an element that maximize the feerate in the following collection.\{U \text{ topological} \mid inc \subseteq U \subseteq exc^c \}. If C \subseteq pot is topological, then C\subseteq B.
Proof. By contraddiction, suppose that there is B like in the hypothesis and C topological, C\subseteq pot and C \not \subseteq B. Since C is topological, possibly replacing t with some of its ancestors, it is no loss to assume anc(t) \subseteq B. Since t\in pot, by 2. of Lemma above, feerate(t) > feerate(pot). By characterization of pot in 1. of Lemma above, we have that feerate(pot) \ge feerate(B). So, feerate(t)> feerate(B). Consider B' = \{t\} \cup B. Clearly, feerate(B')> feerate(B); moreover, since anc(t) \subseteq (B), B' is topological. This is against the definition of B.
Let me know what do you think.