Separation Properties Preserved under Topological Product

Theorem
Let $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$ be a sequence of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$.

$\left({X, \vartheta}\right)$ has one of the following properties iff each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property:


 * $T_0$ (Kolmogorov) Property


 * $T_1$ (Fréchet) Property


 * $T_2$ (Hausdorff) Property


 * $T_{2 \frac 1 2}$ (Urysohn) Property


 * Regular Property


 * Completely Regular Property

If $\left({X, \vartheta}\right)$ has one of the following properties then each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property:


 * Normal Property


 * Completely Normal Property

but the converse does not necessarily hold.

Corollary
$\left({X, \vartheta}\right)$ has one of the following properties iff each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property:


 * $T_3$ Property


 * $T_{3 \frac 1 2}$ (Tychonoff) Property

If $\left({X, \vartheta}\right)$ has one of the following properties then each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property:


 * $T_4$ Property


 * $T_5$ Property

but the converse does not necessarily hold.