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If a normal T2 space is a normal T1 space and a perfectly normal T2 space is a perfectly normal T1 space, then I would assume people will be more likely to use the latter term because T1 is weaker than T2. Are there any historical reasons?

It confuses me because in math, when we are defining an object with a list of axioms, we usually want to use the weakest possible axioms that do not overlap too much.

dodo
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  • A Fréchet space is usually understood as a completely metrizable topological vector space. I can't imagine that this is your intention. – Paul Frost May 17 '22 at 22:44
  • @PaulFrost Oh so people originally use "Hausdorff" here to avoid ambiguity? – dodo May 18 '22 at 06:22
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    I do not know the historical reasons for saying "normal $T_2$" instead of "normal $T_1$". But see https://en.wikipedia.org/wiki/History_of_the_separation_axioms. Moreover I found this wiki-article saying that $T_1$-spaces are also called Fréchet or Tikhonov. I have never heard this in that context, but it shows that mathematical notation is not really standardized. – Paul Frost May 18 '22 at 07:48
  • we usually want to use the weakest possible axioms that do not overlap too much --- There are two competing issues in this, one of which is to have fewer conditions that need to be verified and the other of which is to best convey the most significant properties that arise in practice. Possibly the second of these issues is most relevant here. I'm sure you've seen situations where there is a standard definition of something, but it's known that some parts of the definition can be dropped (e.g. same one-sided identity/inverse in groups). – Dave L. Renfro May 18 '22 at 09:20

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