It is well known that not every closed subspace of a Banach space admits an associated idempotent operator.
Now suppose that we have a filtration $(S_i)_{i\in\mathbb{Z}}$ of closed subsets of a Banach space $B$, such that for all $i$, $S_i\subseteq S_{i+1}$, and suppose moreover that for all $i$, the subspace $S_i$ admits an idempotent operator $P_i$.
Can we conclude that the closed subspaces $\bigcap_i S_i$ and $\mathrm{Closure}(\bigcup_i S_i)$ admit idempotent operators?