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\label{prop:lobes-clopen}
Let $p_0$ denote the node (wedge point) of~$\mathbb{L}$.
The two lobes are topologically complementary
\textbf{clopen} subsets of $\mathbb{L} \setminus \{p_0\}$:
\begin{enumerate}
    \item[\textup{(i)}]
          $L_+ := S^1_\gamma \setminus \{p_0\}$
          and
          $L_- := S^1_\eta \setminus \{p_0\}$
          are each open and closed
          in $\mathbb{L} \setminus \{p_0\}$.
    \item[\textup{(ii)}]
          $L_+ \cap L_- = \varnothing$
          and $L_+ \cup L_- = \mathbb{L} \setminus \{p_0\}$.
    \item[\textup{(iii)}]
          Each lobe is the profinite limit
          of angular sectors:
          $L_+ = \bigcap_{k \geq 1}
          \overline{S^+_k(\textup{B-dom})}$
          and
          $L_- = \bigcap_{k \geq 1}
          \overline{S^-_k(\textup{C-dom})}$.
\end{enumerate}

\emph{Clauses~(i)--(ii).}
The space $\mathbb{L} \setminus \{p_0\}$
is the disjoint union of two punctured circles.
Each is connected, hence both open and closed
in the disjoint-union topology.
Disjointness holds because the two circles
share only~$p_0$, which has been removed.

\emph{Clause~(iii).}
At stage~$k$, the angular sector $S^+_k(b)$
contains points with $B$-coordinate equal to $b$.
The B-dominant boundary points are characterized
by eventual dominance of $B_k$ over $C_k$
(I.T05).
The profinite limit over all stages
selects precisely the B-dominant omega-germs,
constituting the $e_+$-lobe.
The C-lobe argument is symmetric.