Separation Properties Preserved under Topological Product

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


Then $T$ has one of the following properties if and only if each of $\struct {S_i, \tau_i}$ has the same property:

$T_0$ (Kolmogorov) Property
$T_1$ (Fréchet) Property
$T_2$ (Hausdorff) Property
$T_{2 \frac 1 2}$ (Completely Hausdorff) Property
$T_3$ Property
$T_{3 \frac 1 2}$ Property


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

$T_4$ Property
$T_5$ Property

but the converse does not necessarily hold.


Corollary

$T = \struct {S, \tau}$ has one of the following properties if and only if 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

Product Space is $T_0$ iff Factor Spaces are $T_0$
Product Space is $T_1$ iff Factor Spaces are $T_1$
Product Space is $T_2$ iff Factor Spaces are $T_2$
Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff
Product Space is $T_3$ iff Factor Spaces are $T_3$
Product Space is $T_{3 \frac 1 2}$ iff Factor Spaces are $T_{3 \frac 1 2}$


Factor Spaces are $T_4$ if Product Space is $T_4$
Factor Spaces are $T_5$ if Product Space is $T_5$

$\blacksquare$


Sources