Wikipedia states the hierarchy of separation axioms as:

$$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \underset{\text{(Urysohn)}}{T_{2½}} \impliedby \underset{\text{(Regular)}}{T_3} \impliedby \underset{\text{(Tychonoff)}}{T_{3½}} \impliedby \underset{\text{(Normal)}}{T_4} \impliedby \underset{\text{(Completely normal)}}{T_5} \impliedby \underset{\text{(Perfectly normal)}}{T_6} $$

Why doesn't anyone label another property as a "T"? I could add space properties like this:

$$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(US)}}{T_{1¼}} \impliedby \underset{\text{(Weakly Hausdorff)}}{T_{1½}} \impliedby \underset{\text{(KC)}}{T_{1¾}} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \underset{\text{(Urysohn)}}{T_{2½}} \impliedby \underset{\text{(Regular)}}{T_3} \impliedby \underset{\text{(Tychonoff)}}{T_{3½}} \impliedby \underset{\text{(Normal)}}{T_4} \impliedby \underset{\text{(Completely normal)}}{T_5} \impliedby \underset{\text{(Perfectly normal)}}{T_6} \impliedby \underset{\text{(Metrizable)}}{T_7} \impliedby \underset{\text{(Completely metrizable)}}{T_8} \impliedby \underset{\text{(Discrete)}}{T_\infty} $$

Is this acceptable? In particular, why doesn't metrizability qualify as a separation axiom?


1 Answer 1


Your hierarchy is correct, but the question why nobody has introduced $T_x$-names for the additional properties cannot be reasonably answered. It is a question of mathematical tradition, and you cannot exclude someone will suggest to introduce another $T_x$-name, but it is uncertain whether this will be accepted in the mathematical community.

Metrizability has relations to separation axioms, but in my opinion it does not add a new quality in separating points and closed sets. Being completely metrizable definitely has no connection to separation.

  • $\begingroup$ At least discrete spaces being $T_\infty$ is completely reasonable, I think. It literally separates every point from another. $\endgroup$ Dec 10, 2019 at 12:54
  • 1
    $\begingroup$ @DannyuNDos: On the other hand, there is exactly one topology per set that fulfils that condition, namely the discrete topology. I'm not sure if it makes sense to define a separation axiom for just one topology. Otherwise, why not also add $T_{-\infty} = \text{not indiscrete}$? $\endgroup$
    – celtschk
    Dec 10, 2019 at 16:11
  • $\begingroup$ @celtschk $T_\infty$ must imply every other $T_x$. That prevents compact metrizability being $T_9$. $\endgroup$ Dec 11, 2019 at 1:26
  • $\begingroup$ @celtschk Nontriviality as a seperation axiom seems kinda off, for the empty space and the one-point space are always trivial. $\endgroup$ Apr 20, 2020 at 10:23

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