# What is the link (if there is one) between separability and the separation axioms?

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

• Think of a two point set with the indiscrete topology and of an uncountable set with the discrete topology. – David Mitra Aug 12 '12 at 17:46
• 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. – t.b. Aug 13 '12 at 2:54

• "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. – Qiaochu Yuan Aug 12 '12 at 17:37