Separable spaces in topology

Question

I have a few questions about separable space, as I'm trying to understand the meaning of the space having a countable dense subset:

1.Are separable spaces somehow linked to the set of real numbers (e.g. in terms of limitation of size)? 2.What properties do separable spaces have? 3.What purpose have separable spaces been used in the past in topology?

Any help will be appreciated.

Best regards,

Janne

Answer 1

A separable "metric" space has at most cardinality = c. (ie, cardinality of R)