Corollary 1

If $N_\iota$ is closed in $E_\iota$ for every $\iota \in I$, then $N = \sum_\iota N_\iota$ in closed in $E$.

For, the canonical mapping $p_\iota : E \to E_\iota$ is continuous (II, § 4.5, prop. 6) for every $\iota \in I$, hence $p_\iota^{-1}(N_\iota)$ is closed in $E$, and thus the same is true of the intersection $N = \bigcap_{\iota \in I} p_\iota^{-1}(N_\iota)$.