Corollary 2

If the spaces $E_i$ are Hausdorff, then in order that $\mathscr{T}$ be Hausdorff, it is necessary and sufficient that, for every $x \neq 0$ in E, there should exist an index $i \in I$, such that $f_i(x) \neq 0$.

For $\phi(E)$ is then a Hausdorff space, and in order that $\mathscr{T}$ be Hausdorff, it is evidently necessary and sufficient that $\phi$ be injective; note that we can then identify E (with $\mathscr{T}$) with the subspace $\phi(E)$ of $\prod_{i \in I} E_i$ by the mapping $\phi$.