# Intuition for the term “separable” in separable metric space

A metric space is separable if and only if it has a countable dense set.

I find this name unintuitive because I can't see how the above definition relates to the notion of separating a space. There is a connection to Hausdorff spaces, which intuitively can be regarded as separated or separable in a way: A separable Hausdorff space has cardinality at most $2^{\mathfrak c}$. This doesn't help me very much, though.

So why are such spaces called separable?

## 1 Answer

This question has been asked in the past. The term was invented by Maurice Fréchet (1906). But there is no information on why he chose that name.

All we got when we asked the question was wild speculations by modern mathematicians, who had no evidence for their guesses.

Examples of such discussions:

mathoverflow: https://mathoverflow.net/a/16015/454

• Not very satisfying, but thank you very much for looking up these links! – Lukas Kofler May 13 '18 at 20:21