Product Space is T2 iff Factor Spaces are T2

Theorem
Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$.

Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.

Then $T$ is a $T_2$ (Hausdorff) space each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.

Necessary Condition
This is shown in Factor Spaces of Hausdorff Product Space are Hausdorff.

Sufficient Condition
This is shown in Product of Hausdorff Factor Spaces is Hausdorff:General Result.