Theorem: Sequence of Implications of Separation Axioms

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:
{|
|-
| align="center" | ||
| align="center" | ||
| align="center" | Perfectly Normal ||
| align="center" | $\implies$ ||
| align="center" | Perfectly $T_4$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$||
|-
| align="center" | ||
| align="center" | ||
| align="center" | Completely Normal ||
| align="center" | $\implies$ ||
| align="center" | $T_5$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | Normal ||
| align="center" | $\implies$ ||
| align="center" | $T_4$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | ||
|-
| align="center" | $T_{3 \frac 1 2}$ ||
| align="center" | $\impliedby$ ||
| align="center" | Completely Regular ||
| align="center" | $\implies$ ||
| align="center" | Urysohn ||
|-
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | $T_3$ ||
| align="center" | $\impliedby$ ||
| align="center" | Regular ||
| align="center" | $\implies$ ||
| align="center" | $T_{2 \frac 1 2}$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | Semiregular ||
| align="center" | $\implies$ ||
| align="center" | $T_2$ (Hausdorff) ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $T_1$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $\Big \Downarrow$ ||
|-
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | ||
| align="center" | $T_0$ ||
|}

Proof:
The relevant justifications are listed as follows:
* Perfectly Normal implies Perfectly $T_4$ by definition.
* Perfectly $T_4$ Space is $T_5$
* Perfectly Normal Space is Completely Normal.
* Completely Normal implies $T_5$ by definition.
* Completely Normal Space is Normal.
* $T_5$ space is $T_4$.
* Normal implies $T_4$ by definition.
* Normal Space is Completely Regular.
* Completely Regular implies $T_{3 \frac 1 2}$ by definition.
* Completely Regular Space is Regular.
* $T_{3 \frac 1 2}$ Space is $T_3$.
* Regular implies $T_3$ by definition.
* Completely Regular Space is Urysohn.
* Urysohn Space is $T_{2 \frac 1 2}$.
* Regular Space is $T_{2 \frac 1 2}$.
* Regular Space is Semiregular.
* $T_{2 \frac 1 2}$ Space is $T_2$.
* Semiregular implies $T_2$ by definition.
* $T_2$ Space is $T_1$.
* $T_1$ Space is $T_0$.
{{qed}}