Separation Properties Preserved under Topological Product/Corollary

Theorem
Let $\mathbb S = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle T = \struct {S, \tau} = \prod_{i \mathop \in I} \struct {S_i, \tau_i}$ be the product space of $\mathbb S$.

$T = \struct {S, \tau}$ has one of the following properties each of $\struct {S_i, \tau_i}$ has the same property:


 * Regular Property


 * Tychonoff (Completely Regular) Property

If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property:


 * Normal Property


 * Completely Normal Property

but the converse does not necessarily hold.

Proof
A regular space is a topological space which is both a $T_0$ (Kolmogorov) space and a $T_3$ space.

Hence from:
 * Product Space is $T_0$ iff Factor Spaces are $T_0$

and
 * Product Space is $T_3$ iff Factor Spaces are $T_3$

it follows that $T$ is a regular space each of $\struct {S_i, \tau_i}$ is a regular space.

A Tychonoff (completely regular) space is a topological space which is both a $T_0$ (Kolmogorov) space and a $T_{3 \frac 1 2}$ space.

Hence from:
 * Product Space is $T_0$ iff Factor Spaces are $T_0$

and
 * Product Space is $T_{3 \frac 1 2}$ iff Factor Spaces are $T_{3 \frac 1 2}$

it follows that $T$ is a Tychonoff space each of $\struct {S_i, \tau_i}$ is a Tychonoff space.

A normal space is a topological space which is both a $T_1$ (Fréchet) space and a $T_4$ space.

Hence from:
 * Product Space is $T_1$ iff Factor Spaces are $T_1$

and
 * Factor Spaces are $T_4$ if Product Space is $T_4$

it follows that if $T$ is a normal space then each of $\struct {S_i, \tau_i}$ is a normal space.

A completely normal space is a topological space which is both a $T_1$ (Fréchet) space and a $T_5$ space.

Hence from:
 * Product Space is $T_1$ iff Factor Spaces are $T_1$

and
 * Factor Spaces are $T_5$ if Product Space is $T_5$

it follows that if $T$ is a completely normal space then each of $\struct {S_i, \tau_i}$ is a completely normal space.