I was wondering about this topic. Is there a connection between the $T_n$ separation axioms and separability itself?

  • $\begingroup$ Think of a two point set with the indiscrete topology and of an uncountable set with the discrete topology. $\endgroup$ – David Mitra Aug 12 '12 at 17:46
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    $\begingroup$ It'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 for a review). The term separability goes back to Fréchet, see here and here for some historical background. $\endgroup$ – t.b. Aug 13 '12 at 2:54

No; there is no real connection between the two notions. There are both separable and non-separable spaces with any of the separation axioms.

  • $\begingroup$ "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. $\endgroup$ – Qiaochu Yuan Aug 12 '12 at 17:37
  • $\begingroup$ @Qiaochu: While there may be a diachronic connection, synchronically there is none. However, I’ve rephrased it. $\endgroup$ – Brian M. Scott Aug 12 '12 at 17:39

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