2
$\begingroup$

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

  • 0
    Think of a two point set with the indiscrete topology and of an uncountable set with the discrete topology.2012-08-12
  • 1
    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](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 1

4

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

  • 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
  • 0
    @Qiaochu: While there may be a diachronic connection, synchronically there is none. However, I’ve rephrased it.2012-08-12