# Sequence of Implications of Separation Axioms

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## 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:

Perfectly Normal | $\implies$ | Perfectly $T_4$ | $\implies$ | $T_4$ | |||||||||

$\Big\Downarrow$ | |||||||||||||

Completely Normal | $\implies$ | $T_5$ | |||||||||||

$\Big\Downarrow$ | $\Big\Downarrow$ | ||||||||||||

Normal | $\implies$ | $T_4$ | |||||||||||

$\Big\Downarrow$ | |||||||||||||

$T_{3 \frac 1 2}$ | $\impliedby$ | Completely Regular (Tychonoff) | $\implies$ | Urysohn | |||||||||

$\Big\Downarrow$ | $\Big\Downarrow$ | $\Big\Downarrow$ | |||||||||||

$T_3$ | $\impliedby$ | Regular | $\implies$ | $T_{2 \frac 1 2}$ (Completely Hausdorff) | |||||||||

$\Big\Downarrow$ | $\Big\Downarrow$ | ||||||||||||

Semiregular | $\implies$ | $T_2$ (Hausdorff) | |||||||||||

$\Big\Downarrow$ | |||||||||||||

$T_1$ (Fréchet) | |||||||||||||

$\Big\Downarrow$ | |||||||||||||

$T_0$ (Kolmogorov) |

## Proof

The relevant justifications are listed as follows:

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

- Completely Normal implies $T_5$ by definition.

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

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

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Additional Separation Properties