Sequence of Implications of Separation Axioms

Theorem
Let $P_1$ and $P_2$ be separation axioms and let:
 * $P_1 \implies P_2$

mean:
 * If a topological space $T$ satsifies separation axiom $P_1$, then $T$ also satisfies separation axiom $P_2$.

Then the following sequence of separation axioms holds:

Proof
The relevant justifications are listed as follows:


 * Perfectly Normal implies Perfectly $T_4$ implies $T_4$ by definition.


 * Perfectly Normal Space is Completely Normal Space.


 * Completely Normal implies $T_5$ by definition.


 * Completely Normal Space is Normal Space.


 * $T_5$ space is $T_4$ space.


 * Normal implies $T_4$ by definition.


 * Normal Space is Tychonoff (Completely Regular) Space.


 * Completely Regular (Tychonoff) implies $T_{3 \frac 1 2}$ by definition.


 * Completely Regular (Tychonoff) Space is Regular Space.


 * $T_{3 \frac 1 2}$ Space is $T_3$ Space.


 * Regular implies $T_3$ by definition.


 * Completely Regular (Tychonoff) Space is Urysohn Space.


 * Urysohn Space is Completely Hausdorff Space.


 * Regular Space is Completely Hausdorff Space.


 * Regular Space is Semiregular Space.


 * Completely Hausdorff Space is $T_2$ (Hausdorff) Space.


 * Semiregular implies $T_2$ (Hausdorff) by definition.


 * $T_2$ (Hausdorff) Space is $T_1$ (Fréchet) Space.


 * $T_1$ (Fréchet) Space is $T_0$ (Kolmogorov) Space.