I was wondering about this topic. Is there a connection between the $T_n$ separation axioms and separability itself?
What is the link (if there is one) between separability and the separation axioms?
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general-topology
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0Think of a two point set with the indiscrete topology and of an uncountable set with the discrete topology. – 2012-08-12
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1It's probably clear why the $T_n$-axioms are called separation axioms (T stands for German *Trennung* -- "separation"; these axioms go back to the topology book of Alexandroff-Hopf; see [here](http://projecteuclid.org/euclid.bams/1183499379) for a review). The term *separability* goes back to Fréchet, see [here](http://mathoverflow.net/questions/51494) and [here](http://math.stackexchange.com/q/63793/5363) for some historical background. – 2012-08-13
1 Answers
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No; there is no real connection between the two notions. There are both separable and non-separable spaces with any of the separation axioms.
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0"Entirely accidental" is a bit strong. They are both named for an intuitive connection to separating things: in the case of separability it is the idea that you can separate two points of $\mathbb{R}$ by a rational number and in the case of separation axioms it is the idea that you can separate two points by open sets, etc. – 2012-08-12
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0@Qiaochu: While there may be a diachronic connection, synchronically there is none. However, I’ve rephrased it. – 2012-08-12