# 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:

 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:

$\blacksquare$