Corollary 2

If each $E_\iota$ is Hausdorff, so also is $E$ and each $E_\iota$ is closed in $E$.

To prove the first statement apply cor. 1 taking $N_\iota = \{0\}$ for all $\iota \in I$; for the second apply cor. 1 with $N_\iota = E_\iota$ and $N_\kappa = \{0\}$ for every $\kappa \neq \iota$.

We shall show in III, p. 21, cor. 2 that if the $E_\iota$ are Hausdorff and complete then so is their topological direct sum $E$.