# Product Space is T2 iff Factor Spaces are T2

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

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

## Proof

### Necessary Condition

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

$\Box$

### Sufficient Condition

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

$\blacksquare$