Sequence of Implications of Separation Axioms

From ProofWiki
Jump to: navigation, search


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:

$\blacksquare$


Sources