What qualifies as a "separation axiom?"

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?

• What do you mean with “US” (your $T_{1¼}$)? BTW, in trying to search for it, I've come across the following PDF that might interest you: Separation Axioms Between $T_0$ and $T_1$ Dec 11, 2019 at 5:22
• @celtschk Here. topospaces.subwiki.org/wiki/US-space Dec 11, 2019 at 8:36
• Thank you. I didn't know that site; bookmarked it now. Dec 11, 2019 at 9:01

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.
• At least discrete spaces being $T_\infty$ is completely reasonable, I think. It literally separates every point from another. Dec 10, 2019 at 12:54
• @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 $T_{-\infty} = \text{not indiscrete}$? Dec 10, 2019 at 16:11
• @celtschk $T_\infty$ must imply every other $T_x$. That prevents compact metrizability being $T_9$. Dec 11, 2019 at 1:26