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 $\ds T = \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ 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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces